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Dimensionless Numbers in Heat Transfer

It is almost impossible to read an article or listen to a lecture on heat transfer without hearing names like Reynolds. Nusselt, Rayleigh, etc. These names refer to very specific dimensionless numbers that are used to characterize and classify the heat transfer problems. This article attempts to explain the meaning and significance of these numbers and help you to get used to them.

But first, why do we need dimensionless numbers anyway? Well, we actually don't need them but they are useful tools. The nature itself does not have a clue about these numbers. It is not like the air says to itself " boy, my Reynolds number is exceeding 2500 and I am in a pipe so I better switch to my turbulent mode or all the fluid dynamics textbooks will be wrong". We have invented dimensionless numbers to be able to take our knowledge from experimenting with one system to learning about another system with different dimensions. If I have come up with some neat formula for calculating the pressure drop in a 2 inch pipe, can I use that formula for a 4 inch pipe? In a way, we are trying to get rid of dimensions in order to extend our knowledge beyond its source of acquisition. Mr. Osborne Reynolds experimented with pipes of different diameters and discovered that, regardless of the pipe diameter, if the ratio of UD/ exceeds 2500 or so, the flow no longer stays nice and laminar. This ratio is what we call Reynolds number and is probably the most commonly used dimensionless group in fluid dynamics.

Dimensionless numbers allow us to experiment with model cars, airplanes and ships and predict the behavior of the big thing under actual conditions. All we have to establish is to make sure that there is similarity between the model and the actual thing. But, this is beyond the scope of this article.

The Dimensionless numbers we will describe in this article are the most common numbers used in heat transfer:

1. Reynolds Number
2. Nusselt Number
3. Prandtl Number
4. Grashof Number
5. Rayleigh Number

Before getting into the definitions of these numbers, we should define the physical properties of fluids since they show up all over the place in the dimensionless numbers.

Density	Mass of fluid contained in a unit volume. Its units are Kg/m^3 or slugs/ft^3. Typical values: Water = 1000 kg/m^3, Mercury = 13546 kg/m^3, Air = 1.23 kg/m^3, Paraffin Oil = 800 kg/m^3. (at pressure =1.013e+5 Pascals and Temperature = 288.15 K.)
Viscosity	Viscosity, , is the property of a fluid, due to cohesion and interaction between molecules, which offers resistance to sheer deformation of the fluid. Different fluids deform at different rates under the same shear forces. Fluid with a high viscosity such as syrup, deforms more slowly than fluid with a low viscosity such as water. All fluids are viscous, "Newtonian Fluids" obey the linear relationship given by Newton's law of viscosity. , where is the shear stress. is the "coefficient of dynamic viscosity" - The Coefficient of Dynamic Viscosity, , is defined as the shear force, per unit area, (or shear stress ), required to drag one layer of fluid with unit velocity past another layer a unit distance away. Units: Newton seconds per square meter, or Kilograms per meter per second,. (Although note that is often expressed in Poise, P, where 10 P = 1 .) Typical values: Water =1.14xe-3 , Air =1.78e-5 , Mercury =1.552 , Paraffin Oil =1.9 . Kinematic Viscosity, , is defined as the ratio of dynamic viscosity to mass density, . Units: square meters per second, (Although note that is often expressed in Stokes, St, where St = 1e-4 .) Dimensions: . Typical values: Water =1.14e-6 , Air =1.46e-5 , Mercury =1.145e-4 , Paraffin Oil =2.375e-3 . (source:http://www.efm.leeds.ac.uk/CIVE/CIVE1400/Section1/Fluid_properties.htm)
Thermal Conductivity	Thermal conductivity is a measure of the ability of a material to conduct heat. It is defined using the Fourier's law of condution which, relates the rate of heat transfer by conduction to the temperature gradient: where k is the thermal conductivity. Using the Fourier's law we can define the thermal conductivity as the rate of heat transfer through a unit thickness of a material per unit area and per unit temperature difference. A good conductor of heat has a high value of thermal conductivity. The thermal conductivity is expressed in the units of (energy rate/(length.Temperature). In metric system, its unit is W/m.K. Thermal conductivity of most material vary with temperature. For example: T (K) Copper Aluminum 100 482 302 200 413 237 300 401 237 400 393 240 600 379 231 800 366 218   For both cases the thermal conductivity decreases with temperature. Thermal conductivity of most liquids decrease with increasing temperature. Water is, however, an exception to this rule. According to the kinetic theory of gases, the thermal conductivity of gases is proportional to the square root of the absolute temperature and inversely proportional to the square root of the molar mass. It is obvious that the thermal conductivity of a gas increases with the increasing temperature.
Specific Heat	Specific heat is the amount of heat that is required to raise the temperature of a unit mass of a substance by one degree. In a constant pressure process The units for the specific heat are kJ/kg.K (or C). Typical values of Cp for various materials (at 300 K) are shown below: Material Cp (kJ/kg.K) Aluminum (pure) 903 Copper (pure) 385 Gold 129 Silicon 712 Water 4180 Air 1005
Coefficient of Thermal Expansion	This property is usually denoted by and is defined as the change in the density of a substance as a function of temperature at constant pressure. It can be approximated as: In other words, to find the change in density as a function of a change in temperature, we just multiply the density by .
Thermal Diffusivity	When a temperature gradient is applied to a martial, the heat travels from the high temperature region to the low temperature. A measure of how heat propagates through a medium may be defined by the ratio of the heat conducted through the material to the heat stored in the material. Heat capacity is defined as the product of density and specific heat, . The thermal diffusivity is defined as: The thermal diffusivity is, therefore, the ratio of heat conducted through the material to the heat stored per unit volume. The larger the thermal diffusivity the faster the propagation of heat into the material. If the thermal diffusivity is small it means that a big part of the heat is absorbed by the material and only a small portion is conducted through. Some typical value of thermal diffusivity: Material (m^2/s) Aluminum (pure) 97.5e-6 Copper (pure) 113e-6 Gold 127e-6 Glass 0.34e-6 Water 0.14e-6 Air 22.1e-6

We are ready now to explore these numbers in more detail.

Reynolds Number

Reynolds number defined as (where L is a characteristic length) may be interpreted as the ratio of two forces that influence the behavior of fluid flow in the boundary layer. These two forces are the inertia forces and viscous forces:

When the Reynolds number is large, the inertia forces are in command. Viscous forces dominate the boundary layer when the Reynolds number is small. Now, how does this relate to transition from laminar to turbulent flow?

Any real flow of fluid contains small disturbances that will grow given enough opportunities. as long as the viscous forces dominate these disturbances are under control. As the inertia forces get bigger, the viscosity can no longer maintain order and these tiny disturbances grow into trouble makers and we transition to turbulent flow.

Another important quantity of the boundary layer that is influenced by the Reynolds number is its thickness. As the Reynolds number increases, the viscous layer gets squeezed into a smaller distance from the surface.

The value of Reynolds number beyond which the flow is no longer considered laminar is called the critical Reynolds number. For flow over a flat plate, the critical Reynolds number is observed to vary between 1e+5 to 3e+6 depending on the turbulence level in the free stream and the roughness of the surface. We normally use 5e+5 as the critical Reynolds number for flow over flat plates.

Calculation of the Reynolds number is easy as long as you:

• Identify the characteristic length
• Pick the right velocity
• Use a consistent set of units

For flow over a flat plate, the characteristic length is the length of the plate and the characteristic velocity is the free stream velocity. For pipes the characteristic length is the pipe diameter and the characteristic velocity is the average velocity through the pipe obtained by dividing the volumetric flow rate by the cross-sectional area (we are assuming that the pipe is full, of course). For pipes with a non-circular cross-section, the characteristic length is the Hydraulic Diameter defined as 4A/P, where A is the cross-sectional area of the duct and P is the wetted perimeter. You can easily verify that for a circular pipe the hydraulic diameter equals the pipe diameter. For non-cicular pipes the average velocity is used as the characteristic velocity. The situation gets messy when you are dealing with a problem that has many velocity and length scales such as the flow inside a computer cabinet. You must decide, based on your design objectives, which length and velocity length scales make sense for calculation of the Reynolds number.

Nusselt Number

Nusselt number is the dimensionless heat transfer coefficient and appears when you are dealing with convection. It, therefore, provides a measure of the convection heat transfer at the surface. It is defined as hL/k where, h is the heat transfer coefficient, L is a characteristic length and k is the thermal conductivity. But, what does this grouping mean from a physical standpoint? Let's find out.

I am afraid that we have to look at the boundary layer in order to explain the concept of Nusselt number. We will, of course, cover the basics of the boundary layer in a separate tutorial but for now it suffices to say that when a fluid flows over a solid surface, the first layer of the fluid stick to the boundary (we even have a name for this thing called, no slip condition). This causes the flow to retard in the vicinity of the wall. As we move away from the wall the effect of this no slip thing gets smaller and smaller up to a point where it is no longer felt by the fluid. To get to this point, though, we have had to go through a layer of fluid who still knows about the wall. This layer is called the boundary layer. This was the effect of the wall on the velocity (or momentum). A similar argument applies when, for example, a cold fluid flows over a hot surface. The first layer of the fluid (which is now stuck to the surface) gets its heat from the surface through pure conduction. It then gives its newly acquired energy to all of the other fluid molecules that it comes in contact with as they pass by it (this is convection). As we move further and further away from the wall, the effect of the hot wall is felt less and less (it, of course, depends on the thermal conductivity of the fluid). Eventually, there comes a point where the fluid does not have a clue about the hot wall. The layer of fluid between the wall and this point is called the thermal boundary layer. It is where all of the action is taking place (as far as heat transfer between the solid and fluid is concerned). Before continuing with the Nusselt number, let us define another dimensionless property:

Prandtl Number

Heat transfer gurus have invented another dimensionless number called the Prandtl number which is a grouping of the properties of the fluid but it has a significance to our discussion. Prandtl number is defined as:

It is the ratio of momentum diffusivity (kinematic viscosity) to thermal diffusivity. It can be related to the thickness of the thermal and velocity boundary layers. It is actually the ratio of velocity boundary layer to thermal boundary layer. When Pr=1, the boundary layers coincide. Typical values of the Prandtl number are:

When Pr is small, it means that heat diffuses very quickly compared to the velocity (momentum). This means the thickness of the thermal boundary layer is much bigger than the velocity boundary layer for liquid metals.

Now, back to the Nusselt number.

Remember that we explained how the first fluid layer stick to the solid surface and the heat is transferred via conduction. Well, let's write this equation:

In a boundary layer situation the characteristic length is the thickness of the boundary layer.

Consider a fluid layer of thickness L and a temperature difference of across this layer. Heat transfer by convection can be calculated as hwhile heat transfer by conduction is k/L. Dividing the convection heat transfer to the conduction heat transfer, we get:

So, the Nusselt number may be viewed as the ratio of convection to conduction for a layer of fluid. If Nu=1, we have pure conduction. Higher values of Nusselt mean that the heat transfer is enhanced by convection.

Grashof Number

You see this number and you should think of natural or free convection. The Grashof number is the ratio of buoyancy forces to the viscous forces.

In natural convection the Grashof number plays the same role the is played by the Reynolds number in forced convection. The buoyant forces are fighting with viscous forces and at some point they overcome the viscous forces and the flow is no longer nice and laminar. For a vertical plate, the flow transitions to turbulent around a Grashof number of 10^9.

Rayleigh Number

The Rayleigh number is the product of Grashof and Prandtl numbers. It turns out that in natural convection the Nusselt number scales with Rayleigh rather than just Grashof. Most correlations in natural convection are of the form:

where .

References: For writing this note we have used material from two excellent heat transfer text books:

1- Heat Transfer, A Practical Approach, Yunus A. Dengel, Mcgraw-Hills

2- Fundamentals of Heat and Mass Transfer, Frank Incropera and david DeWitt, 4th Edition, John Wiley& Sons