These non-dimensional numbers are helpful tools in heat transfer. Reynolds number, gives the information, whether the flow is inertial or viscous force dominant. It tells us whether the flow is laminar or turbulent. Another eg: is, say you need to study the aerodynamics of an air craft. Thus you have two options (1) build the actual air craft, instrument it and test run for study (expensive, complicated and not practical). Or (2) you can develop a scaled down model and study it in a wind tunnel, provided the same Re is generated in the wind tunnel. Therefore using wind tunnel study would be the simple and professional way. This wouldn't have been possible is the concept of Re was not there.

Nusselt number (Nu) on the other hand is an non dimensional heat transfer coefficient. It gives the comparison between the conduction and convection heat transfer rates.

Prandtl (Pr) number gives the information about the type of fluid. Also it provides the 

information about the thickness of thermal and hydrodynamic boundary layer.

also, there is another number called the 'j' factor, which is the function of Nu, Re and Pr. J factor is commonly used in heat transfer studies.

Hope it helped you !

Regards,

Rajesh. 

These numbers (dimensionless in nature) are  very useful in evaluation of:

1. The comparison between the conduction and convection heat transfer rates.

2. The information about the type of fluid (lamina, turbulent, etc.) .

3. Provides the information about the thickness of boundary layer (for thermal and hydrodynamic).

Actually by using or you can say practicing these numbers, one can easily guess (theoretically) about important properties and mechanism behind heat transfer, fluid flow, etc. in a particular system under study and hence, are very essential and important for Thermal Engineering and related subjects for both Academic and R&D activities.

These numbers (dimensionless in nature) are  very useful in evaluation of:

1. The comparison between the conduction and convection heat transfer rates.

2. The information about the type of fluid (lamina, turbulent, etc.) .

3. Provides the information about the thickness of boundary layer (for thermal and hydrodynamic).

Actually by using or you can say practicing these numbers, one can easily guess (theoretically) about important properties and mechanism behind heat transfer, fluid flow, etc. in a particular system under study and hence, are very essential and important for Thermal Engineering and related subjects for both Academic and R&D activities.

Возможно, не решая уравнений, объединить физические величины в безразмерные комплексы и получить вид безразмерных (критериальных) уравнений с меньшим числом переменных. Решение этих уравнений позволяет находить искомые величины. Точные критериальные уравнения отыскиваются путем проведения соответствующих экспериментов. Примером безразмерного критерия подобия является критерий (число) Рейнольдса

Этот критерий определяет характер движения: для ламинарного движения в трубе Rе l-I04 (промежуточные значения относятся к неустойчивому движению).

Другим важным критерием является  критерий Нуссельта, который имеет вид, аналогичный критерию Био, разница заключается в том, что коэффициент теплопроводности в первом случае берется для газов, а во втором для обтекаемого тела.

Объединение характерных величин в безразмерные критерии позволяет определить количественные соотношения множества явлений.

Thanks to all. Now I've gotten my answer.

Reynolds = Inertial Force / Viscous Force = rho U L / mu

Nusselt = Covective Heat Transfer / Conductive Heat Transfer = h L / k

Prandtl = Momentum Diffusivity / Thermal Diffusivity = nu / alpha = Cp mu / k

Reynolds determines if the flow is laminar or turbulent.

Nusselt calculates the transference of heat relative to pure conduction.

Prandtl represents how big is the viscous boundary layer relative to thermal boundary layer.

Peclet = Reynolds * Prandtl

Nusselt = F ( Reynolds , Prandtl )             F = Function

See:

https://www.academia.edu/11945927/An%C3%A1lisis_Dimensional

and its references.

I have to express dismay that such a question was even raised. Answers are there in every basic book on Heat Transfer, if the teacher (or research supervisor) is unable to clarify. Can't we use the resources offered by Research Gate in a better way!

The above is the answer I gave another on RG. It applies equally well here!

What is the difference between Nusselt number and Biot number?? - ResearchGate. Available from: https://www.researchgate.net/post/What_is_the_difference_between_Nusselt_number_and_Biot_number/1 [accessed Jul 15, 2015].

 I respect your opinion. But let me say you something. How many great discoveries have emerged from naive questions. I surprise the answers you may find to these questions here in RG. See for example:

Can Bernoulli's equation properly be derived from the first principle of thermodynamics?

You can see many of the participants should read the basic text book again or at first time. Others are more profound or philosophical in their answers.

But if you are tolerant, maybe you can learn from others. Many information flow in this discussions. Some irrelevant, some interesting. Maybe you make some friends or fans. Many follow the discussions only to see what other people is saying about without participation. Let the people express themselves, I would say. We do not dictate the policy of RG. Otherwise omit them.

I agree with Andres Granados. The Point of asking question is not to simply get right answers, but to interact with different minds to know about their way of thinking. Even before asking this question i consulted the books and googled about these numbers, but it is the (good/bad) habit of mine, whenever i learn a new thing i ask questions about it, this gives me the different prospects of the same topic. I am new in this field and i started to note these dimensionless numbers have so much importance in heat transfer problems so i asked to get an answers from experience researchers.

@Andres and Zeeshan,

You are both right. As Andres says, there are sometimes irrelevant answers (and sometimes even wrong answers). My worry was the poser of the question (or readers) being misled. RG has no mechanism to ensure quality except for upvotes and downvotes, which in my opinion is not enough.

You might perhaps have noticed that, when I answer they are extensive and complete. Often that is followed by some insipid responses that makes me think that my answer has not been read at all!  If I were to give a 'proper' and 'complete' answer to the above question, it would have needed pages and would merely be a repetition. That was the reason for my dismay. My apologies.

To complete Dr Ayesha Sohail good answer I would say that using of nondimensional parameters (P) reduces by three or four  (D) the number of total variables (V) which the problem is depended on. According to Dimensional analysis by PI theorem V-D = P

V = Variables of the problem (dimensional) V_i

D = Dimensions of the variables in  basic units used (mass-length-time-temperature or force-length-time-temperature) U_j

P = Parameters (nondimensional) P_k

See for example:

https://www.academia.edu/11945927/An%C3%A1lisis_Dimensional

or

https://www.academia.edu/11942882/Mec%C3%A1nica_y_Termodin%C3%A1mica_de_Sistemas_Materiales_Continuos    (appendix E, section 1.2)

(If you don't understand spanish read the references in there. Any good text book of fluid mechanics or heat transfer has this PI (or Buckingham) theorem explained in basic terms. The following corollary is also important)

The Nusselt, Reynolds and Prandtl numbers are dimensionless numbers presented in the form of a ratio that compares two phenomena.

- The Nusselt number (Nu=h*L / kf) is the ratio of convection to pure conduction heat transfer, where kf is the conductivity of the fluid.

- The Reynolds number (Re=v*L / {kinematic viscosity}) is the ratio of the inertia and viscous forces.

- The Prandtl number (Pr={kinematic viscosity} / {thermal diffusivity}) is the ratio of the momentum and thermal diffusivities.

As the surface heat transfer coefficient h depends on several parameters, these dimensionless numbers are used in order to provide empirical correlations taking into account all the influencing variables (surface geometry, fluid thermal properties, fluid velocity,...)

In the case of forced convection, several correlations relating the Nusselt number to the Reynolds and the Prandtl numbers are provided in the literature.

In the case of free convection, the Nusselt number is related to the Grashof number which measures the ratio of buoyancy forces to viscous forces and plays nearly the same role as the Reynolds number in forced convection.

Grashoff Number = rho^2 g L^3 beta delta T / mu^2

beta = coefficient of volume expansion at constant pressure VI.3.1.2 in my book.

From the book "Fundamentals of Aerodynamics" by Anderson (which i highly recommend):

"On a physical basis, the Prandtl number is an index which is proportional to the ratio of energy dissipated by friction to the energy transported by thermal conduction"

A good compilation of nondimensional parameters

Land, N. S.

A Compilation of Nondimensional Numbers.

NASA Report No. SP-274, National Aeronautics and Space Administration (Washington), 1972.

Also see these

A Dictionary of Scientific Units Including Dimensionless Numbers and Scales (5ed)_-_H. G. Jerrard-D. B. McNeill.

Units Dimensions and Dimensionless Numbers_-_David Carl Ipsen.

And my brief article

https://www.academia.edu/11945927/An%C3%A1lisis_Dimensional

A good catalogue of dimensionless numbers:

Dimensionless Physical Quantities in Science and Engineering_-_Josef Kunes

what will be the effect by increasing the "Pr" on heat transfer

as we all know that by Increasing Pr velocity decreases

so it means that the Heat transfer will be reduced.

Am i right r wrong?

Prandtl number compares two molecular phenomena of the same medium (fluid). The first (which is the nominator of Pr) is how fast and easy the kinetic energy (or the momentum) can be transferred between the molecules of the fluid and the characteristic property is the viscosity. The second one (the denominator of Pr) says how fast and easy the heat is transferred between the same molecules and the characteristic property is the thermal conductivity.

As an example imagine a cold plate upon which there is a hot flow with some velocity. In the area close to the plate two boundary layers will be formed one with respect to velocity and one for temperature. The one of the velocity will start with a zero value at the surface (non-slip condition) and then the velocity will start increasing as we go far from plate till it matches the velocity of the hot flow. Similar the temperature boundary layer starts with the temperature of the plate at the surface and then it starts increasing till it reaches the temperature of the hot flow far from the plate.

If viscosity is large the velocity boundary layer will be big, cause it will take time for the flow near the plate to adjust to the main hot flow. If thermal conductivity is low the heat has difficulty to travel so the temperature boundary layer will be bigger.

Now the Prandtl number compares these two boundary layers. Bigger velocity boundary layer means higher Pr number (more energy is lost to overcome viscosity), while bigger temperature boundary layer means lower Pr number (more energy is lost to overcome conductivity).

Graphs, mainly on a logarithmic scale, help in determining the form of the criterion equations. If you get a straight line on a log-log scale (Nu vs Re), you can be sure that the relationship is Nu = CReA in relation to the Reynolds number. Worse if it is not a linear relationship, because then we usually start with the equation in the form of y = ao + a1x1 + a11x12 + ....., and then e.g. stepwise regression method. Apart from the guidelines regarding the model's form, you will get nothing more from the graph. It is only the form of the equation, mainly in the form of a power monomial, along with the value of its parameters that may suggest, what type of heat exchange you have to deal with. Of course, the influence of other factors should be taken into account: Gr number (natural convection), µ/µw, D/L etc. Regards,

These are dimesionless numbers used in fluid flow/heat transfer analysis. It helps to compare differnt properties/ parameters

Nusselt number is the ratio of convection heat transfer to pure conduction heat transfer, (K- Thermal conductivity of fluid)

Reynolds number is the ratio of the inertia to the viscous forces avting on a fluid.

Prandtl number is the ratio of the momentum and thermal diffusivities.

A dimensionless number should be freed from any physical dimension. It does not matter if the flow takes place in a gas pipeline or in a micro-channel we can analyse by comparison all its properties by simply considering dimensionless quantities. Convective heat transfer could be hence derived analytically through dimensional analysis. But for several engineering situations as flow in internal and external pipes, arnd plates in laminar and turbulent regimes, convective heat transfer coefficient h is proportional to Nusselt number which in turn deends on Re and Pr via correlations Nu=aRebPrc where a, b and c are estimated from the experiments. This approach is probably not "elegant " since we did not yet determine the physical sens of these constants (It is as if you correlate the weight with mass through W=9.8 m), but this approach is very helpful for the engineers !

Yes, I agree with Samir Farhat ,

Most of convective heat transfer correlation was obtained based on dimensional analysis. Unfortunately, the heat transfer rate is closed related to the geometry of the heat transfer surface (the shape, dimensions, orientation, surface roughness, boundary conditions, etc), which can not be formulated in dimensional analysis. Thus, most of the correlation are very simple. Hence, the uncertainty of such a correlation will be in question.

Thank you.

I. In the application of dimensional analysis to transport phenomena (engineering), dimensionless quantities are often correlated by means of empirical correlations that often follow (or closely follow) power laws. Power laws possess the attribute of scale invariance. From that follows a straight-line log–log plot, what often helps to identify a power law trend. Power laws often provide fairly good correlations with a minimal number of parameters.

II. For instance; for forced convection, we may quite generally express the Nusselt number as a function of both the Reynolds and the Prandtl numbers for great diversity of conditions, namely geometric, with simply 3 parameters: Nu = k·Rep·Prq. It may be even possible to further reduce the number of degrees of freedom by means of a predetermined constant (possibly the unit: p + q = cte. = 1), to impart a compensated 'weight' to each contribution. It may be convenient to take the parameters as time varied.

III. This very limited number of degrees of freedom may contribute to stabilize numerical procedures, namely those related with real-time parameter estimation, with real-time optimization, and with adaptive control; particularly if processes occurs concurrently, and if no prior 'learning phase' is reserved for just parameter estimation. Imposed (limited) disturbances may also play a stabilizing role. Both real-time optimization and with adaptive control may be suited to benefit from concurrent parameter estimation.

IV. The least-squares fitted parameters obtained after the logarithmic-linearized correlation are biased for the exponential correlation, but typically close to the unbiased estimates if obtained directly from the original power-law correlation.

V. The recursive least squares algorithm (RLS) allows for (real-time) dynamical application of least squares (LS) regression to a time series of time-stamped continuously acquired data points. As with LS, there may be several correlation equations with the corresponding set of dependent (observed) variables. RLS is the recursive application of the well-known LS regression algorithm, so that each new data point is taken in account to modify (correct) a previous estimate of the parameters from some linear (or linearized) correlation thought to model the observed system. For RLS with forgetting factor (RLS-FF), acquired data is weighted according to its age, with increased weight given to the most recent data. No prior 'learning phase' is required.

VI. Application example for comparable power law correlation ― While investigating adaptive control and energetic optimization of aerobic fermenters, I have applied the RLS-FF algorithm to estimate the parameters from the KLa correlation, used to predict the O2 gas-liquid mass-transfer, while giving increased weight to most recent data. Estimates were improved by imposing sinusoidal disturbance to air flow and agitation speed (manipulated variables). The proposed (adaptive) control algorithm compared favourably with PID. The power dissipated by agitation was accessed by a torque meter (pilot plant). Simulations assessed the effect of numerically generated white Gaussian noise (2-sigma truncated) and of first order delay. This investigation was reported at (MSc Thesis):

Thesis Controlo do Oxigénio Dissolvido em Fermentadores para Minimi...

it has significant importance