1. Presenting 10 dimensionless numbers along with their importance By Khair Muhammad 16CH05

2.  1 Biot Number  2 Nusselt Number  3 Peclet Number  4 Graetz Number  5 Stanton Number  6 Grashof Number  7 Fourier Number  8 Rayleigh Number  9 Prandtl Number  10 Lewis Number

3. The ratio of the conductive heat resistance within an object to the convective heat transfer resistance across that object's boundary Mathematically Bi = Conductive resistance of Solid/ Convective resistance of Fluid = L/kA/1/hA = hL/k L = Volume of body/ Surface area of body

4.  This number is used in transient heat transfer and used in extended surface heat transfer calculations.  This ratio determines whether or not the temperatures inside a body will vary significantly in space, while the body heats or cools over time, from a thermal gradient applied to its surface.  Greater the number more non uniform will be the temperature field.

5. Example  The physical significance of Biot number can be understood by imagining the heat flow from a small hot metal sphere suddenly immersed in a pool, to the surrounding fluid. The heat flow experiences two resistances: the first within the solid metal (which is influenced by both the size and composition of the sphere), and the second at the surface of the sphere. If the thermal resistance of the fluid/sphere interface exceeds that thermal resistance offered by the interior of the metal sphere, the Biot number will be less than one.

6.  Having a Biot number smaller than 0.1 labels a substance as "thermally thin," and temperature can be assumed to be constant throughout the material's volume. The opposite is also true: A Biot number greater than 0.1 (a "thermally thick" substance) indicates that one cannot make this assumption, and more complicated heat transfer equations for "transient heat conduction" will be required to describe the time-varying and non-spatially-uniform temperature field within the material body

7.  It is the ratio of convective to conductive heat transfer across (normal to) the boundary Mathematically  Nu= Thermal Resistance due to conduction in fluid/ Thermal Resistance due to convection in fluid = L/kA/1/hA = hL/k or hx/k or hD/k  L is the characteristic length  The Nusselt number greater than 1 indicates that the resistance due to conduction is higher than that due to convection OR q(convection) is greater than q(conductive)  So the movement of fluid(s) will result in more heat transfer  When Nusselt number is less than 1 than the situation is opposite to that of above.  Note: Nu occurs on liquid surface while Bi on solid one.

8.  Used to know that whether the natural convection is efficient or not.  The physical interpretation of Nusselt number is the enhancement of heat transfer due to convection over conduction alone. If Nu=1, then, than your fluid is stationary and all heat transfer is by conduction. With Nu>1, the fluid motion enhances heat transfer by advection.  The Nusselt number is of paramount importance because it contains the heat transfer coefficient information. Indeed, one interpretation of Nusselt number is simply that of dimensionless heat transfer coefficient. In simplest terms, Nu is simply a dimensionless heat transfer coefficient.

9.  It is the ratio of the thermal energy convected to the fluid to the thermal energy conducted within the fluid.  Or Pe= Lv/D  L is characteristic length  v is velocity  D is Dispersion or Diffusion 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  There are various forms of this number.  This is relevant to the study of transport phenomena in a continum.  For mass transfer, it is defined as  Pe= Re. Sc:  For heat transfer, the Péclet number is defined as:  Pe= Re.Pr  This number is called Brenner number for particulate motion

10.  The Peclet number is a measure of the relative importance of advection versus diffusion, where a large number indicates an advectively dominated distribution, and a small number indicates a diffuse flow.  Also Pe= υ LcharD =convection transport into diffusion transport  tells about the boundary layer characteristics of the flow, now related to the fluxes of energy  Advection is the movement of some material dissolved or suspended in the fluid.  So if you have pure water and you heat it you will get convection of the water. You can't have advection because there is nothing dissolved or suspended in the fluid to advect  While convection is the movement of a fluid, typically in response to heat.

11.  Gz= (Heat transfer by convection/Heat transfer by conduction)* D/L  = (D/L)* Re*Pr  or  =Pe* D/L  Here D is hydraulic diameter of tubes or any cross sectional substance  This number characterizes the Laminar flow in a conduit (Channel).  This is number is used for the heat flow to the fluid through circular pipes.  This number is useful in determining the thermally developing flow entrance length in ducts. A Graetz number of approximately 1000 or less is the point at which flow would be considered thermally fully developed.[2]

12.  When used in connection with mass transfer the Prandtl number is replaced by the Schmidt number, Sc, which expresses the ratio of the momentum diffusivity to the mass diffusivity.  Another form of Gz number  Gz = UD2/kx  where U is the velocity of the fluid, D the diameter of the pipe, κ the fluid thermal diffusivity ( λ / ρ cp) and x the axial distance along the pipe.  Gz represents the ratio of the time taken by heat to diffuse radially into the fluid by conduction  The inverse of Gz number represents the entrance effects of laminar flow in a pipe.  This number is useful in determining the thermally developing flow entrance length in ducts. A Graetz number of approximately 1000 or less is the point at which flowwould be considered thermally fully developed  The Graetz problem is a fundamental tube flow problem that couples fluid flowwith heat and/or mass transfer. It is critically important in dealing with chemicalreactors, heat exchangers, blood flow and a host of other phenomena. A niceaspect of it is that there is an analytical solution to the problem for laminar flow, both developing and fully developed

13.  Stanton number is proportional to { (heat transfered) / (thermal capacity of fluid) } and is used in heat transfer in general and forced convection calculations in particular. It is equivalent to (Nu / (Re.Pr)). It is normally defined in one of the following forms :  St= Wall heat transfer/ Heat transfer by convection  Mathematically ( for heat transfer)  St= Nu/Pe or Nu/ Pr.Re or  St= h/Cp. Rho. V  Or St= h/Cp. G  Here G is mass velocity  For Mass Transfer  St= Sh/ Re.Sc or  St= beta/u  Where beta is mass transfer coefficient

14.  The Stanton number arises in the consideration of the geometric similarity of the momentum boundary layer and the thermal boundary layer, where it can be used to express a relationship between the shear force at the wall (due to viscous drag) and the total heat transfer at the wall (due to thermal diffusivity).  The Stanton number is a useful measure of the rate of change of the thermal energy deficit (or excess) in the boundary layer due to heat transfer from a planar surface Heat and Mass transfer, Forced convection studies  For most of the convection problems, the skin friction coefficient has been calculated or tabulated in heat transfer data book and so it has known for us  So if you know the value of skin friction coefficient Cf, we can calculate Stanton number by Reynolds analogy  St = Cf/2 where St=h/¶Vcp(Std definition) So hence we can find the value of heat transfer coefficient, h by h =¶VCp x Cf/2 ¶ - density Cp-specific heat capacity So we don't have to find Nu, Re, Pr to know the 'h' value and calculation become easy

15.  the Grashof number is a ratio of buoyant forces to viscous forces acting on a fluid  So  Gr = Inertial forces * Buoyancy forces / (Viscous forces)2  Gr= B3 (rho)2 deltaT. rho. g / (meu)2  The buoyant force. When an object is placed in a fluid, the fluid exerts an upward force we call the buoyant force.  Gr number values much higher than one, indicate the negligence of viscous forces

16.  used in analyzing the velocity distribution in free convection systems. In the systems encountered most commonly in BSL, free convection is is the natural tendency of a substance to migrate due to some driving force. In free convection, the driving force is a buoyancy force caused by a temperature gradient, as the fluid would be at rest in the absence of temperature variations. The Grashof number is analogous to the Reynolds number in forced convection

17.  The Fourier Number (Fo) is a dimensionless group which arises naturally from the non dimensionalization of the conduction equation.  Fo= Rate of heat transfer across in volume / Rate of heat storage in volume  Or  Fo= Diffusive transport rate/ Storage rate  Mathematically  Fo= alpha* t /L2  Where alpha is the thermal diffusivity = k/Cp.P

18.  It 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.  It is very widely used in the description and prediction of the temperature response of materials undergoing transient conductive heating or cooling.  characterizes transient heat conduction

19.  The Rayleigh number is defined as the product of the Grashof number, which describes the relationship between buoyancy and viscosity within a fluid, and the Prandtl number,  Mathematically  Ra= Gr*Pr  where g is acceleration due to gravity, β ' is coefficient of thermal expansion of the fluid, Δ T is temperature difference, x is length, ν is kinematic viscosity and κ is thermal diffusivity of the fluid. Gr is the Grashof Number and Pr is the Prandtl Number.

20.  The magnitude of the Rayleigh number is a good indication as to whether the natural convection boundary layer is laminar or turbulent.  the Rayleigh number is below a critical value for that fluid, heat transfer is primarily in the form of conduction; when it exceeds the critical value, heat transfer is primarily in the form of convection.  In geophysics, the Rayleigh number is of fundamental importance: it indicates the presence and strength of convection within a fluid body such as the Earth's mantle. The mantle is a solid that behaves as a fluid over geological time scales

21.  : the ratio of the product of the coefficient of viscosity and the specific heat at constant pressure to the thermal conductivity in fluid flow used especially in the study of heat transfer in mechanical devices Or  the ratio of momentum diffusivity to thermal diffusivity.  Pr = v / α  Or  the ratio of the fluid viscosity to the thermal conductivity of a substance, a low number indicating high convection.

22.  Fluids with small Prandtl numbers are free-flowing liquids with high thermal conductivity and are therefore a good choice for heat conducting liquids.  liquid metals are very good heat transfer liquids. Interestingly, air is a decent heat transfer liquid as well, whereas typical organic solvents are not. With increasing viscosity, the momentum transport dominates over the heat transport, which makes these liquids a bad choice for heat conduction.  Usually, the Prandtl number is assumed to be around 0.7 for gases and around 6.9 for water.  In heat transfer problems, the Prandtl number controls the relative thickness of the momentum and thermal boundary layers. When Pr is small, it means that the heat diffuses quickly compared to the velocity (momentum). The mass transfer analog of the Prandtl number is the Schmidt number.

23.  The ratio of thermal diffusivity to mass diffusivity.  The Lewis number is therefore a measure of the relative thermal and concentration boundary layer thicknesses. The Lewis number can also be expressed in terms of the Prandtl number and the Schmidt number as Le = Sc / Pr.  The Lewis number is a material constant

24.  It is used to characterize fluid flows where there is simultaneous heat and mass transfer.  one of the most important numbers for studying combustion

25.  There are various dimensionless numbers and each of them has one or more forms.  These numbers are very important for the engineers and professionals with mathematical relatd backgrounds  These 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 you need to solve a problem numerically, dimensionless groups help you to scale your problem.