Importance and Significance of Various Dimensionless Numbers in Different Modes of Heat Transfer

 Introduction


Dimensionless numbers are used in heat transfer to measure the importance of different physical factors and phenomena. These numbers are often used to simplify complex equations and to compare multiple phenomena in different scenarios. The most common dimensionless numbers in heat transfer are the Biot number, the Prandtl number, the Nusselt number, the Grashof number, and the Rayleigh number.

These dimensionless numbers are important in heat transfer applications because they allow engineers to compare different systems and determine the relative importance of the various physical factors. For example, if the Biot number is low, it indicates that conduction is the dominant form of heat transfer, while a high Biot number indicates that convection is the dominant form of heat transfer. 

In conclusion, dimensionless numbers are important tools for heat transfer engineers. They allow engineers to compare the relative importance of physical phenomena in different systems, and they can also be used to predict the performance of a system. Various Dimensionless numbers in conduction and radiation heat transfer phenomenons are mentioned below.


Boltzman number 


Boltzmann constant, (symbol k), a fundamental constant of physics occurring in nearly every statistical formulation of both classical and quantum physics. The constant is named after Ludwig Boltzmann, a 19th-century Austrian physicist, who substantially contributed to the foundation and development of statistical mechanics, a branch of theoretical physics. Having dimensions of energy per degree of temperature, the Boltzmann constant has a defined value of 1.380649 × 10−23 joule per kelvin (K), or 1.380649 × 10−16 erg per kelvin. The molar gas constant R is defined as Avogadro’s number times the Boltzmann constant

The physical significance of k is that it provides a measure of the amount of energy (i.e., heat) corresponding to the random thermal motions of the particles making up a substance. For a classical system at equilibrium at temperature T, the average energy per degree of freedom is kT/2. In the simplest example of a gas consisting of N noninteracting atoms, each atom has three translational degrees of freedom (it can move in the x-, y-, or z-directions), and so the total thermal energy of the gas is 3NkT/2.


Biot number :

Biot number is the ratio of internal conductive resistance within the body to the external convective resistance at the surface of the body.

Bi=internal Conductive Resistance/Extensive Convective Resistance


‘L’is the characteristic length of the body.

‘K’ is the thermal conductivity of the body.

 ‘h’ is the convection of the body.

This number gives the relation between internal resistance and external resistance for heat transfer. This equation is recognized by the name of French physicist  Jean Baptiste Biot.



The biot number has the following significances:-

  •  The biot number shows the relation between the internal conductive resistance and convective resistance at the surface of the object.

  •  It is used as a criterion for lumped system analysis.

  •  The higher value of the biot number indicates that the internal resistance in the object is higher than external resistance.

  •  The lower value of the biot number indicates that the conductive resistance of an object is comparatively lower than external resistance.

  •  Biot number along with Fourier numbeer helps to analyze objects by lumped system analysis.




Graetz number

Graetz number is the reciprocal of Fourier number with time replaced by x/U applicable mainly to transient heat conduction in laminar pipe flow.
The quantity is named after the physicist Leo Graetz.
The Graetz problem is a fundamental tube flow problem that couples fluid flow with heat and/or mass transfer.  It is critically important in dealing with chemical reactors, heat exchangers, blood flow and a host of other phenomena.  A nice aspect of it is that there is an analytical solution to the problem for laminar flow, both developing and fully developed.  An analytical solution for turbulent flow does not yet exist, but we can investigate turbulence effects in Comsol.

           The analytical solution for fully developed hydrodynamics and developing thermal profiles depends upon the Graetz number that defines the decay rate of the initial inlet temperature.


 Formula :


It is also expressed as :

Where, 

di is the internal diameter in round tubes or hydraulic diameter in arbitrary cross-section ducts

L is the length

Re is the Reynolds number and

Pr is the Prandtl number.


We can therefore conclude that the Graetz Number is the reciprocal of the Fourier number and that it can also be related to the Reynolds number and the Prandtl number.


Fourier number :


In physics and engineering, the Fourier number (Fo) or Fourier modulus, named after Joseph Fourier, is a dimensionless number that characterizes transient heat conduction. Conceptually, it is the ratio of diffusive or conductive transport rate to the quantity storage rate, where the quantity may be either heat (thermal energy) or matter (particles). The number derives from non-dimensionalization of the heat equation (also known as Fourier's Law) or Fick's second law and is used along with the Biot number to analyze time dependent transport phenomena.

Fourier number is the dimensionless quantity used in the calculation of unsteady-state heat transfer. The Fourier number is the ratio of the rate of heat conduction to the rate of heat stored in a body. It is derived from the non-dimensionalization of the heat conduction equation. 

The Fourier number for heat transfer is given by,

Where , 

α = Thermal diffusivity 

𝜏 = Time (Second)

Lc = Characteristics length …….( Lc = volume / area )

Fourier number significance :


The significances of the Fourier number are as follows:-

  • The Fourier number indicates the relation between the rate of heat conduction through the body and the rate of heat stored in the body.

  • The larger value of the fourier number indicates the higher rate of heat transfer through the body.

  • The larger value of the fourier number indicates the lower rate of heat transfer through the body.

 

Brinkman Number :

The Brinkman number (Br) is a dimensionless number related to heat conduction from a wall to a flowing viscous fluid, commonly used in polymer processing. It is named after the Dutch mathematician and physicist Henri Brinkman. There are several definitions; one is

  

where

• μ is the dynamic viscosity;

• u is the flow velocity;

• κ is the thermal conductivity;

• T0 is the bulk fluid temperature;

• Tw is the wall temperature;

• Pr is the Prandtl number

• Ec is the Eckert number

It is the ratio between heat produced by viscous dissipation and heat transported by molecular conduction. i.e., the ratio of viscous heat generation to external heating. The higher its value, the slower the conduction of heat produced by viscous dissipation and hence the larger the temperature rise. 

In, for example, a screw extruder, the energy supplied to the polymer melt comes primarily from two sources:

• viscous heat generated by shear between elements of the flowing liquid moving at different velocities;

• direct heat conduction from the wall of the extruder.

The former is supplied by the motor turning the screw, the latter by heaters. The Brinkman number is a measure of the ratio of the two.

 

Nusselt Number :

The Nusselt number is the ratio of convective to conductive heat transfer across a boundary. The convection and conduction heat flows are parallel to each other and to the surface normal of the boundary surface, and are all perpendicular to the mean fluid flow in the simple case.

where h [W/m2.K] is the convective heat transfer coefficient of the flow, L is the characteristic length, and k [W/m.K] is the thermal conductivity of the fluid.

• Selection of the characteristic length should be in the direction of growth (or thickness) of the boundary layer; some examples of characteristic length are: the outer diameter of a cylinder in (external) cross flow (perpendicular to the cylinder axis), the length of a vertical plate undergoing natural convection, or the diameter of a sphere. For complex shapes, the length may be defined as the volume of the fluid body divided by the surface area.

The thermal conductivity of the fluid is typically (but not always) evaluated at the film temperature, which for engineering purposes may be calculated as the mean-average of the bulk fluid temperature and wall surface temperature.

In contrast to the definition given above, known as average Nusselt number, the local Nusselt number is defined by taking the length to be the distance from the surface boundary to the local point of interest.

The significances of Nusselt number are as follows:-

1] The Nusselt number gives the relation between heat transfer by conduction and heat transfer by convection.

2] When Nu = 1, It means that the rate of heat transfer by conduction is equal to the rate of heat transfer by convection or we can say that fluid is stationary.

3] If the Nu > 1, Then the heat transfer by convection is greater than heat transferred by convection.

 

 

 






References

[1] Cengel, Yunus A. (2002). Heat and mass transfer operation (2nd ed.). McGraw-Hill.

[2] The Nusselt number ; Whiting School of Engineering. Retrieved 3 April 2019

[3] Özisik, M. N. (1973) Radiative Transfer and Interactions with Conduction and Convection, J. Wiley & Sons, New York, London, Sydney, Toronto.

[4] Siegel, R. and Howell, J. R. (1972) Thermal Radiation Heat Transfer, McGraw Hill Book Comp., New York et al.

[5 ]https://en.wikipedia.org/wiki/Fourier_number

[6] https://en.wikipedia.org/wiki/Graetz_number

[7] Nellis, G., and Klein, S. (2009) "Heat Transfer" (Cambridge).

[8] Shah, R. K., and Sekulic, D. P. (2003) "Fundamentals of Heat Exchanger Design" (John Wiley and Sons).

 

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