Dimensionless Numbers In Heat Transfer

 

In Heat Transfer, Dimensionless numbers are used to characterize and classify heat transfer problems. This blog attempts to explain the meaning and significance of these numbers and help you to get used to them.

Introduction:

Dimensionless numbers are scalar quantities commonly used in fluid mechanics and heat transfer analysis to study the relative strengths of inertial, viscous, thermal, and mass transport forces in a system.

Dimensionless numbers are equal for dynamically similar systems, systems with the same geometry, and boundary conditions. This makes them a powerful tool for scaling operations from model to pilot and beyond. It helps in determining a systematic arrangement of the variables in the physical relationship.

Importance of Dimensionless Number:

We can experiment with model automobiles, planes, and ships and make predictions about how the big thing will behave in real-world scenarios thanks to dimensionless numbers. All that needs to be done is to confirm that the model and the real thing are similar.

Why Dimensionless Numbers used in Heat Transfer:

•         Dimensionless numbers allow for comparisons between very different systems.
•         Dimensionless numbers tell you how the system will behave.
•      Many useful relationships exist between dimensionless numbers that tell you how specific things influence the system.
•         Dimensionless numbers allow you to solve a problem more easily.
•       When we need to solve a problem numerically, dimensionless groups help you to scale your problem.

 

1.     Reynolds Number:

It is defined as the ratio of the inertia force to viscous force.

            

 

where

·       U flow Speed

·       L characteristic length

·       υ kinematic viscosity

 

Significance:            

 ·      Reynolds number signifies the relative importance of the inertia to the viscous forces occurring in the flow systems.

·    The higher the value of Re the greater will be the relative contribution of inertia effect. The smaller the value of Re, the greater will be the relative magnitude of the viscous stresses.

·     Reynolds number is taken as an important criterion of kinematic and dynamic similarities in forced convection heat transfer.

 

 

2.   Prandtl Number:

It is named after Ludwig Prandtl, who introduced the concept of boundary layer in 1904 and made significant contributions to boundary layer theory. Prandtl Number is the ratio of kinematic viscosity to thermal diffusivity of the fluid. It tells about relative importance between the momentum diffusivity to the thermal diffusivity.

 

 

where

• υ is momentum diffusivity (m²/s)
• α is thermal diffusivity (m²/s)
• μ is dynamic viscosity (N.s/m²)
• K is thermal conductivity (W/m. k)
• c_p is specific heat (J/kg. k)
• ρ is density (kg/m³)

Significance:

• Prandtl number provides a measure of the relative effectiveness of the momentum and energy transport by diffusion.
• Prandtl number is a connecting link between the velocity field and temperature field, and its value strongly influences relative growth of velocity and thermal boundary layers.
• Solely on fluid and its characteristics.
• It is the proportion of a fluid’s momentum to heat diffusivity.
• Additionally, it is the proportion of thermal to velocity boundary layers.
• Prandtl number is a connecting link between the velocity field and temperature field, and its value strongly influences relative growth of velocity and thermal boundary layers.
• Prandtl number is dependent on dynamic viscosity. By increasing dynamic viscosity Prandtl number also increases.  

 

3.   Grashof Number: 

Grashof Number indicates the relative dominance of inertia and buoyant forces over viscous force. It is defined as the ratio of the product of inertia force and buoyancy force to the square of viscous force. Grashof number is related with natural convection heat transfer.

Where

• g is the acceleration due to Earth’s gravity,
• β is the coefficient of thermal expansion,
• T_s is the wall temperature
• T_∞ is the bulk temperature,
• L is the vertical length,
• ν is the kinematic viscosity

Significance:

·        The ratio of natural convection’s buoyancy and viscous forces.

·        Grashof number is used in natural convection, the Reynolds number is used in forced convection of fluid flow.

4.   Nusselt Number:

Nu established the relation between convective film coefficient
(h), thermal conductivity of the fluid (K) and a significant length.  It is the ratio of heat flow rate by convection process under a unit temperature gradient to the heat flow rate by conduction process under a unit temperature gradient through a stationary thickness of L metres.

where

• k_f is the thermal conductivity of the fluid [W/m. K]
• L  is the characteristic length
• h is the convective heat transfer coefficient (W/m². K)

Significance:

·     The ratio of the boundary layer’s convective to conductive heat transfer coefficient.

·       Low Nu results in increased conduction and laminar flow.

·       High Nu = more convection = turbulent flow.

·       It can also be seen as the material’s conduction resistance to convection resistance.

·       Nu = f(Re, pr)

·       It represent dimensionless heat transfer coefficient. 

 

5.   Biot Number:

The Biot number is the ratio of the internal resistance of a body to heat conduction to its external resistance to heat convection. Therefore, a small Biot number represents small resistance to heat conduction, and thus small temperature gradients within the body.

where

• k thermal conductivity of the body
• h convective heat transfer coefficient
• L characteristic length (m) (Volume/Total Surface Area)

Significance:

      ·       It is used when there is a transient (unsteady state) heat transfer.

    ·       The ratio of the body’s internal heat transfer resistance to its external heat transfer resistance exterior thermal resistance divided by internal thermal resistance.

      ·       When the body’s internal heat transfer resistance is less than 0.1, conduction occurs more quickly than convection at the surface.

       ·       It is use in unsteady heat transfer. 

6.   Rayleigh Number:

Rayleigh number is the product of the Grashof and Prandtl numbers.

Where

• g is the acceleration due to Earth’s gravity,
• β is the coefficient of thermal expansion,
• T_s is the wall temperature
• T_∞ is the bulk temperature,
• L is the vertical length,
• ν is the kinematic viscosity

·       It number plays an important role in determining whether natural convection flow is laminar or turbulent. 

 

7.    Fourier Number:

The Fourier number is a measure of heat conducted through a body relative to heat stored.

 

where

• α = Thermal diffusivity
• t = Time (in Second)
• L_c = Characteristics length

Significance:

·        The ratio of the rate of heat storage to the rate of heat rate of heat conduction.

·        A large value of the Fourier number indicates faster propagation of heat through a body.

·        Used in conjunction with the Biot number to solve heat transport problems in transient states.

·        The Fourier number for MT is utilized for mass transfer by diffusion.

·        It can also be thought of as the interval from the present to the time required to reach a steady state.

·        It represent dimensionless time 

 

 

Conclusion:

Dimensionless numbers play an important role in analysing fluid dynamics and heat and mass transfer problems. They provide a method by which complex phenomena can be characterized, often by way of a simple, single-number comparison. It gives the relationship of Heat Transfer with Mass Transfer and Fluids Mechanics. 

References

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

2.       Heat and Mass Transfer, R.K. Rajput, S. Chand

3.       A Heat TransferTextbook, by John H. Lienhard IV and John H. Lienhard V

 

Published By:-

Prof. Avadhoot Rajurkar

(54) Swarada Kulkarni

(58) Jayesh Mane

(63) Sakshi Niphade

(76) Aditya Rathod

(77) Darshan Rathod