Dimensionless Numbers and Problem 4.5 - 1 Calotes, Lou Janssen Espiloy, Anna Lorraine Olandria, Cyril Kaye

Dimensionless Numbers Pure numbers without any physical units, it does not change if one alters one's system of units of measurement, for example from English units to metric units.

Simple ratio of two dimensionally equal quantities (simple) or that of dimensionally equal products of quantities in the numerator and in the denominator - Kunes, J. ( 2012 ). Dimensionless Physical Quantities in Science and Engineering. Elsevier

Purposes of Dimensionless Numbers • Allow comparisons for very different systems. • Indicates 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 you need to solve a problem numerically, dimensionless groups help you to scale your problem.

Fields of Application • • • • • • •

Heat and Mass Transfer Optics Astronomy Aerodynamics Mathematics Electronics Chemistry

Bergman, T., Lavine, A., Incropera, F., Dewitt, D. ( 2011 ). Introduction to Heat

Bergman, T., Lavine, A., Incropera, F., Dewitt, D. ( 2011 ). Introduction to Heat

Recall:

Osborne Reynolds 1842 - 1912

=

 

=   = =

Reynolds Number

Important dimensionless quantity in fluid mechanics that is used to help predict flow patterns in different fluid flow situations.  

< 2000 : Laminar Flow = 2100 – 4000 : Transition Flow > 4000 : Turbulent Flow

Péclet Number Jean Claude Péclet 1793 - 1857

 

Defined to be the ratio of the rate of advection of a physical quantity by the flow to the rate of diffusion of the same quantity driven by an appropriate gradient. Reynolds Number counterpart to Where: thermal energy   transfer = =

 

Biot Number Jean Baptiste Biot 1774 - 1862  

Provides a measure of the temperature drop in the solid, relative to the temperature difference between the surface and the fluid during transient processes.  

Bergman, T., Lavine, A., Incropera, F., Dewitt, D. ( 2011 ). Introduction to Heat

Biot Number

Bergman, T., Lavine, A., Incropera, F., Dewitt, D. ( 2011 ). Introduction to Heat

Biot Number

Bergman, T., Lavine, A., Incropera, F., Dewitt, D. ( 2011 ). Introduction to Heat

Biot Number

   

The resistance to conduction within the solid is much less than the resistance to convection across the boundary fluid layer. It is reasonable to assume uniform temperature distribution The temperature difference across the solid is much larger than that between the surface of the solid and the fluid.

Bergman, T., Lavine, A., Incropera, F., Dewitt, D. ( 2011 ). Introduction to Heat

Prandtl Number Ludwig Prandtl (1875 - 1953)

•   Where:

Where:

Typical Range of Prandlt Numbers for Selected Fluids:

Grashof Number Franz Grashof (1826 - 1893)

•   Where:

Where:

Lewis Number Warren Lewis (1882 – 1975)

•   Where:

Where:

Fourier Number Joseph Fourier (1768 - 1830)

•   Where:

Nusselt Number

Wilhelm Nusselt • • • •

A German engineer Born on November 25, 1882 Died on September 1, 1957 Developed dimensional analysis of heat transfer

Nusselt Number •   where

Nusselt Number •   • Gives the ratio of the actual heat transferred between the plates by moving fluid to the heat transfer that would occur by conduction. • Characterize heat flux from a solid surface of a fluid.

Nusselt Number •  Large Nusselt number signifies very efficient convection • Example: For Turbulent,

Nusselt Number •  Free Convection • Forced Convection

Nusselt Number •  Free Convection of a Vertical Wall

Nusselt Number •  Free Convection from Horizontal Plates • top surface of a hot object in a colder environment or bottom surface of a cold object in a hotter environment

Nusselt Number •  bottom surface of a hot object in a colder environment or top surface of a cold object in a hotter environment

Nusselt Number •  Flat Plate in Laminar Flow

Nusselt Number •  Forced Convection in Turbulent Pipe Flow Gnielinski Correlation

  Valid:

Nusselt Number •  Forced Convection in Turbulent Pipe Flow Dittus-Boelter Equation   Valid:

Nusselt Number •  Forced Convection in Turbulent Pipe Flow Sieder-Tate Correlation   Valid:

Nusselt Number •  Forced Convection in Fully Developed Laminar Pipe Flow Nusselt number are constant-valued. For internal flow: where - convective heat transfer coefficient; - hydraulic diameter; - thermal conductivity of the fluid

Nusselt Number •  Forced Convection in Fully Developed Laminar Pipe Flow Convection with uniform surface heat flux for circular tubes Convection with uniform surface temperature for circular tubes

Stanton Number

Stanton Number •  Measures the ratio of heat transferred into a fluid to the thermal capacity of a fluid. • Characterize heat transfer in forced convection flows

• When Pr = 1, Thomas Edward Stanton (1865-1931)

Graetz Number

Leo Graetz • • • •

A German physicist Born on September 26, 1856 Died on November 12, 1941 First to investigate the propagation of electromagnetic energy

Graetz Number •Applicable   to many transient heat conduction in laminar pipe flow

where is the velocity of the fluid, the diameter of the piper, the fluid thermal diffusivity, axial distance along the pipe

Graetz Number  

 

Time taken by heat to diffuse radially into the fluid by conduction, sometimes called “relaxation time”

Graetz Number  

 

Time taken for the fluid to reach distance

Graetz Number •  

Graetz Number •; radial   temperature profiles are fully developed ; larger values thermal boundary layer development has to be taken into

Graetz Number •   where is the diameter in round tubes or hydraulic diameter in arbitrary crosssection ducts, is the length, is the Reynolds number, is the Prandtl number.

Rayleigh Number

Rayleigh Number • • • •

John William Strutt, 3rd Baron Rayleigh Born on November 12, 1842 Died on June 30, 1919 Discovered argon with William Ramsay

Rayleigh Number •  Determines how heat is transferred throughout the fluid • Associated with buoyancy-driven flow (free convection). • When , heat transfer due conduction • When , heat transfer due to convection

Rayleigh Number •    where, – thermal

diffusivity – surface characteristic temperature length --ambient Rayleightemperature number of a characteristic length -– Grashof acceleration number duefor to gravity characteristic length – Prandtl thermal number expansion coefficient – kinematic viscosity

 

If

 

If

Problem 4.5-1 Heating Air by Condensing Steam

Air is flowing through a tube having an inside diameter of 38.1 mm at a velocity of 6.71 m/s, average temperature of 449.6 K, and pressure of 138 kPa. The inside wall temperature is held constant at 477.6 K by steam condensing outside the tube wall. Calculate the heat transfer coefficient for a long tube and the heat-transfer flux.   Given:

 

 

 

   

   

Assumptions: • Air is an ideal gas. • No heat loss • No internal heat generation • Steady flow

Required:

 

Determining Type of Flow •

Determining Reynold’s Number:  

 Average Density of the Flowing Air:

Viscosity of air at the mean bulk Temperature: from the Appendix

Substituting the values into the equation:

For Turbulent Flow The •   heat transfer coefficient is given by: (Equation 4.5-8 on page 261)

The heat-transfer flux is given by:

Values: