Dimensionless number in chemical engineering

DIMENSIONLESS NUMBER USED
IN CHEMICAL ENGINEERING
Common Nomenclature used in Dimensionless Number
1. ρ Density of the fluid = Mass/ Volume
2. u Velocity of the fluid
3. Di Diameter of Pipe
4. μ Viscosity of the fluid (Dynamic or absolute Viscosity)
5. ν Kinematic viscosity = Ratio of the absolute viscosity or dynamic viscosity to the density of the fluid.
Kinematic viscosity is also referred as the MOMENTUM DIFFUSIVITY of the fluid (ability of the
fluid to transport momentum).
6. α Thermal Diffusivity
7. D Mass Diffusivity
8. Cp Specific heat capacity at constant pressure
9. k Thermal conductivity of the fluid
10. Lch Characteristic Length
11. Vch Characteristic Velocity
12. h Heat transfer coefficient
13. K Mass transfer coefficient
14. β Volumetric Thermal Expansion = 1/T = for ideal fluid
15. DH Hydraulic Diameter
16. Ts Surface temperature
17. T∞ Bulk Temperature
18. T Absolute Temperature
19. g Acceleration due to Gravity
20. t Characteristic Time

I. Reynold Number
Reynold Number is defined as the ratio of INERTIA Force to VISCOUS Force.
Inertia Force: - A force acting opposite in direction to an accelerating force acting on a body and equal to
product of accelerating force and mass of body.
Viscous Force: - A force between a body and a fluid [either liquid or gas] moving past it in a direction so as
to oppose the flow of fluid past object.
This both the parameter is convenient variable to identify the flow condition either it is laminar flow or
turbulent flow.
Reynold Number write as,
… … … … …. (1)
Reynolds number can be interpreted as,
➢ Viscous force is dominant, means that flow is slow and all the fluid particle in line. For this condition,
the flow of fluid is laminar and value of Reynolds number is low i.e., Re < 2000.
o For very low value of Reynolds number, viscous creeping motion is created where inertial force
effect is negligible.
o Flow velocity profile for laminar flow in circular pipe is parabolic in nature. {Maximum flow
in centre of the pipe and minimum flow at pipe walls}
o Average velocity = 0.5 * umax
o For Laminar flow
▪ Fluid is moving in straight line
▪ Most favourable condition low velocity and high viscosity
▪ Adjacent layer of the fluid moves to parallel to each other
➢ Inertial force is dominant, means that flow is flowing faster and flow is in irregular movement of
particle. As flow is moving faster, particle produced eddies in the fluid. For this condition, the flow of
fluid is Turbulent and value of Reynolds number is high i.e., Re > 4000.
o Flow velocity profile is fairly flat across centre section of pipe and drops rapidly extremely
close to walls.
o Average velocity ≈ centreline velocity of pipe.
o For Turbulent flow
▪ Fluid is in irregular movement also fluid layers cross each other with high lateral
mixing.
▪ Most favourable condition low viscosity and high velocity.

➢ If the Reynold number is between 2000 < Re < 4000, the flow is called Transient Flow. Transient
flow from laminar to turbulent flow depend on the geometry, surface roughness of pipe, free-stream
velocity, surface temperature and type of the fluid.
Physical Significance of Reynold Number
• Essential for the calculation of friction factor in a fluid mechanics.
• It is used to characterize the relative strength of inertial and viscous forces which in results how the
fluid flow behaves i.e., laminar or turbulent.
II. Prandtl Number
This dimensionless number is most widely used in HEAT TRANSFER & FLUID DYNAMICS.
Prandtl Number is defined as the ratio of the MOMENTUM DIFFUSIVITY to the THERMAL
DIFFUSIVITY.
It is also defined as the ratio of the CONVECTIVE HEAT TRANSFER to the CONDUCTIVE HEAT
TRANSFER.
Momentum Diffusivity (Kinematic Viscosity, 𝑣): - How the fluid is flowing on any surface deal with
convective heat transfer.
- It tells us about material’s resistance to shear- flows in relation to density.
- Ratio of Dynamic viscosity of the fluid to the Density of the fluid is defined as the Momentum
Diffusivity. [ 𝑣 =μ/𝜌 ]
Thermal Diffusivity [𝛼]: - It explain conductive behaviour in heat transfer.
- It is the ratio of Thermal conductivity to the product of Density & Specific Heat capacity.
𝛼=𝑘/(𝑐𝑝⋅𝜌)
Prandtl Number written as,
… … … (2)

As per the equation 2, it is clear that Prandtl Number is:
o Not depend on the surrounding [It means it does not affect by the where fluid is flowing
whether is in tube or pipe].
o Not depend on any characteristic length.
o It is only dependent on fluid properties which type of fluid and fluid state.
- Small value of Prandtl Number, Pr < 1 → Thermal diffusivity dominant it means thermal boundary
layer thickness is more and heat transfer taken by conduction.
- Large value of Prandtl Number, Pr > 1 → Momentum diffusivity dominant it means momentum
boundary layer thickness is more and heat transfer taken by convection.
- If Prandtl number = 1, then momentum diffusivity equals to the thermal diffusivity and mechanism
and rate of heat transfer are similar to those for momentum transfer.
Based on the value of Prandtl number, it can predict that convective or conductive heat transfer take place in
system also Prandtl number control relative thickness of the momentum and thermal boundary layer.
Standard value of Prandtl Number
Important Note: Laminar & Turbulent Prandtl Number
Equation 2 of Prandtl Number, which only showing the laminar part and only valid for laminar flow
For turbulent flow,
Turbulent Prandtl Number is in non-dimensional term, 𝑃𝑟 = 𝜗𝑡/𝛼𝑡
- It is the ratio of momentum eddy diffusivity to heat transfer eddy diffusivity.
It is simply described swirling/ rotation of useful to solved problem of turbulent boundary layer flow.
Simplest model for turbulent Prandtl no is Reynolds Analogy which yield a turbulent Prandtl no 1.
For air at room temperature 0.71
Water at 18 ℃ 7.56
For gas 0.6 to 1
Mercury 0.015
Liquid metal 0.001 to 0.03
Oil 50-2000
Polymeric Liquid Near to 10000

III. Schmidt Number
Schmidt Number is the analogous to the Prandtl Number in Mass transfer calculation.
It is defined as the ratio of Viscous Diffusion rate to Molecular (Mass) diffusion rate that means ratio of
momentum diffusivity to molecular diffusivity.
Schmidt Number = Viscous Diffusion Rate (Momentum Diffusivity) / Mass diffusion rate (Molecular
Diffusivity)
= Fluid boundary layer / mass transfer boundary layer
= 𝑣 / D
Sc = μ / ρ * D … … … … … (3)
Schmidt number is used to characterize fluid flows in which there are simultaneous momentum and mas
diffusion convection processes.
• Schmidt number physically relates relative thickness of hydrodynamic layer and mass transfer
boundary layer.
o If Schmidt Number = 1 → It implies that momentum and mass transfer by diffusion are
comparable, also the velocity and concentration boundary layer almost coincide with each
other.
o If Schmidt number >1 → It implies that thickness of momentum boundary layer at any location
on plate is more than thickness of concentration boundary layer.
o Standard value of Schmidt number → For gas- gas = 0.5 to 2
→ For gas-liquid and liquid-liquid = 100 to 10000
• Schmidt number is dependent on temperature of the fluid because diffusion rate is dependent on the
temperature. Hence, Schmidt number is inversely proportional to the temperature.
Turbulent Schmidt Number: - It is the ratio of turbulent transport of momentum and turbulent transport of
mass that means ratio of eddy viscosity to eddy diffusivity.
Sct = 𝜗𝑡/k

IV.Lewis Number
Lewis number is the ratio of Schmidt number to Prandtl Number.
It is also defined as the ratio of thermal diffusivity to mass diffusivity.
Lewis number = Schmidt Number / Prandtl Number
= [Momentum Diffusivity / Molecular Diffusivity]
[Momentum Diffusivity/Thermal Diffusivity]
= Thermal Diffusivity / Molecular Diffusivity
Le = α = k
D ρ*Cp*D … … … … (4)
• Lewis number is used to characterize fluid flow where there is simultaneous heat and mass transfer
occur.
• It measures of relative thermal and concentration boundary layer thickness.
• A Lewis number of unities [Le =1] indicates that thermal boundary layer and mass transfer by
diffusion are comparable and temperature and concentration boundary layer almost coincide each
other.
V.Peclet Number
Peclet number is the ratio of rate of advection of a physical quantity by the flow to rate of diffusion
[matter or heat] of the same quantity driven by an appropriate gradient.
Peclet number distinguish between Mass Transfer and Heat Transfer.
❖ Peclet number in Mass Transfer
o In mass transfer, Peclet number is the product of Reynolds number and Schmidt Number.
Where, Reynolds number describes flow regime
Schmidt number used to characterize fluid flows in which there are simultaneous
momentum and mass diffusion convection processes.
o Peclet Number in mass transfer is defined as Advection transport rate to the Diffusive
transport rate
o Advection: It is the transport of something from one to another place by bulk motion of fluid
generally horizontally.

Peclet number = PeM = Re * Sc
PeM = u * Lch = u
D D / Lch … … … (5)
[Here, Diameter is replaced by characteristic length of horizontal geometry, Advection process is done
horizontal geometry. For example, cross flow over a cylinder, diameter is chosen but if flow is parallel to
cylinder axis, then length of cylinder is more important.]
Characteristic Length is defined as the ratio of Wetted area to Wetted Perimeter. It is also defined as the
volume of the system divided by its surface.
❖ Peclet number in Heat transfer
o In heat transfer, Peclet number is the product of Reynolds number and Prandtl number.
o It is also defined as the ratio of the thermal energy convected to fluid to the thermal energy
conducted within the fluid.
o Another definition is the ratio of the heat transfer by motion of a fluid to heat transfer by
thermal conduction.
Peclet number, PeH = Re * Pr
PeH = u * Lch
α … … … … (6) [Here, characteristic length is considered and
thermal diffusivity, 𝛼=𝑘/(𝑐𝑝⋅𝜌)]
❖ If pecelt number is small, conduction is important.
VI.Nusselt Number
Nusselt Number is used mainly in calculation of Heat Transfer.
It is closely related to Peclet Number [Both the dimensionless number used to describe ratio of thermal
energy convected to the fluid to thermal energy conducted.]
Nusselt number is equal to the dimensionless temperature gradient at surface and it provides a measure of
convection heat transfer occurring at surface.

Nusselt Number = Convective heat transfer / Conductive heat transfer
= Qconv / Qcond
Nu = h * Lch
k … … … …(7)
- Nusselt number is dependent on two parameters
o It is depended on surrounding it means where the fluid is flowing e.g., on flat surface, fluid
flowing through tube, etc.
o It is also depended on type of fluid dealing with also the characteristic length it means
geometry of the system.
[Important Note: Difference between Prandtl number and Nusselt number is that Prandtl number is
not depend on the surrounding whereas Nusselt number is dependent on surrounding.]
Nusselt number is the function of Reynold Number and Prandtl number.
Nu = 𝒇(𝐑𝐞 , 𝑷𝒓)
Nusselt number represents the enhancement of heat transfer through a fluid layer as a result of convextion
relative to conduction across the same fluudi layers.
Nu = 1 → for a fluid layer represent heat transfer across the layer by pure conduction.
Nu is higher → system is more effective in convection process.
Different relation of Nusselt number for Laminar and Turbulent Flow described as:
A. For Laminar Flow
a. For internal flow (for circular tube)
i. In the fully developed region, both friction factor and heat transfer coefficient
remain constant when flow is laminar in a tube with constant surface tempera
Nu = h * Lch = 3.66 … … (8)
k
ii. For fully developed laminar flow in a circular tube subjected to constant surface
heat flux, Nusselt number is constant
Nu = h * Lch = 4.34 … … (9)
k
b. For external flow (for flat plate at distance x)

i. Relation gives average heat transfer coefficient for the entire flat plate when the
flow is laminar over the entire plate.
… …(10)
Where, Re > 5 * 105
Pr > 0.6
B. For Turbulent Flow
a. External Turbulent flow [for flat plate at a distance x]
i. This realtion gives average heat transfer coefficient for entire flat plate when the
flow is turbulent over plate.
… … (11)
Where, 5 *10 < Rex < 107
0.6 ≤ Pr ≤ 60
b. Internal turbulent flow
i. Dittus Bolter equation
1. For a fully developed (hydrodynamically and thermally ) turbulent flow in
a smooth circular tube, the local Nusselt number obtain by Dittus Bolter
equation.
2. It is less accurate in case of large temperature difference across fluid and
also for rough tubes.
3. This dittus bolter equation may be used for small to moderate temperature
difference , Twall – Tavg. With all properties evaluated at an average
temperature
… …(12)
Where, ReD ≥ 10000
0.6 ≤ Pr ≤ 160
The value of n = 0.3 for fluid being cooled
n = 0.4 for fluid being heated

ii. Sieder -Tate equation
1. When difference between surface and fluid temperature is large, it may be
necessary to account for variation of viscosity with temperature.
2. It applies to normal fluids in turbulent flow in long straight pipe.
… …(13)
Where, ReD ≥ 10000
0.7 ≤ Pr ≤ 16700
𝝁 = fluid viscosity at bulk fluid temperature
𝝁w = fluid viscosity at wall temperature
[Tube wall have difference temperature it shows the relation of temperature with viscosity.]
Significance of Nusselt Number:
- Enhancement of heat transfer through a fluid layer as a result of convection relative to
conduction across the same fluid layer.
- To helping in determination of heat transfer coefficient calculation.
VII. Sherwood Number
Sherwood number is analogous to Nusselt number in the Mass transfer calculation.
Sherwood number is defined as the ratio of Convective mass transfer rate to the Diffusive mass
transfer rate.
It is to the concentration boundary layer.
Sherwood Number = Convective mass transfer rate / Diffusive mass transfer rate
Sh = k * Lch
D … … … … (14)
Sherwood number is the function of Reynolds number & Schmidt Number.
Sh = ƒ (Re , Sc)

- Depending on the geometry (where the fluid is flowing) , relation of Sherwood Number
1.For flat plate at a distance x
▪ For laminar flow
… … … (15)
Where, Rex < 3 * 105
▪ For turbulent flow
… … …(16)
Where, Rex > 3 * 105
2.For a circular tube / pipe flow
▪ For laminar flow
… … … (17)
Where, Re < 10000
dc = diameter of pipe
▪ For turbulent flow
… … … (18)
Where, Re > 10000
3.For sphere [mass transfer from a freely falling liquid droplet]
… … … (19)

Physical significance of Sherwood number:
- Sherwood no in mass transfer analogous to Nusselt no in heat transfer.
- Sherwood number depending on surrounding it means where the fluid flowing such as flate plate, pipe ,
sphere, etc..
- Sherwood number value helps to know which parameter are influenced the system and can find out
favourable condition for system if can not the parameter can tune according to this.
- It helps to determine mass transfer coefficient.
VIII. Stanton Number
Stanton number has relation in terms of heat & mass transfer.
➢ Stanton number for Heat Transfer
o Stanton number is defined as the ratio of heat transfer into a fluid to the thermal capacity
[heat capacity] of fluid.
o It is used to characterize heat transfer in forced convection flows.
o It can also defined as ratio of Nusselt number to Peclet number [Peclet number is the product of
Reynolds number and Prandtl number].
St = Nu = Nu
Pe Re* Pr
… … … … (20)
o Stanton number indicated the degree of amount of heat delivered by fluid when there is transfer
between fluid surface and fluid.
o Stanton number arise in consideration of geometric similar to momentum boundary layer and
thermal boundary layer where it used to express a relationship between shear force at wall and
total heat transfer at the wall due to thermal efficiency.
➢ Stanton number for Mass Transfer
o Stanton number is defined as the ratio of Sherwood number to Peclet number [Peclet number is
the product of Reynolds number and Schmidt Number.].
St = Nu = Sh
Pe Re *Sc
… … … (21)

IX. Biot Number
Biot number is useful in heat transfer calculation.
Biot number is defined as the ratio of Internal conductive resistance to the External convective
resistance.
Biot number = Conduction resistance within the body [Internal resistance]
Convection resistance at the surface [External convective resistance]
Bi = h * Lch
k … … … (22)
[For sphere, consider Lch = Di (diameter of sphere) , heat transfer across the diameter in case of sphere
Here k represent thermal conductivity of the solid]
The value of Biot number is a criterion which gives a direct indication of the relative importance of
conduction and convection in determining the temperature history of a body being heated or cooled by
convection at its surface.
The Biot number tells us about how much the thermal conduction and convection capability for a given solid
object.
If Bi < < 1 → Material has very high thermal resistance by convection [external resistance].
If Bi > 40 → Material has very high thermal resistance by conduction [internal resistance].
[Characteristic Length : The characteristic length in most of relevant problem becomes heat characteristic
length that is ratio of volume of body to the area of heated or cooled surface of body.
L = V / A]
[Major difference between Nusselt number and Biot number is that
In Nusselt number thermal conductivity of the fluid will considered while In Biot number thermal
conductivity of solid will considered in calculation.]

X. Fourier Number
It is used in heat & mass transfer calculation.
It is used to describe and predict the temperature response of the material undergoing transient
conductive heating or cooling.
Fourier Number for Heat Transfer
 Fourier number is the ratio of rate heat conduction to rate of thermal storage.
 It can explain as how the heat is conducting through material and how heat is storing in that material.
 Dimensionless time is defined in Fourier number for a temperature change to occur.
o Fourier number = Conduction rate / rate of thermal storage
… … … (23)
Fourier Number for Mass Transfer
 It is the ratio of Diffusive transport rate to Mass storage rate
o Fourier number = Diffusive transport rate / Mass storage rate
… … … (24)
[Note :- If thermal diffusivity α , is high then Fourier number is high then rate of heat conduction is good in
lighter material then heavier body. Hence, lighter body cool faster than heavier body. ]
Major application of Fourier number is in Lumped system analysis.
Lumped System Analysis: - During transient heat transfer, the T normally varies with time as well as
position. In the special case of variation with time but not with position, the T of the medium change
uniformly with time. Such heat transfer system is called LUMPED SYSTEMS.
It is used in which no temperature gradient exists. This means that the internal resistance of the body
(conduction) is negligible in comparison with the external resistance (convection).

This situation arises when Biot number < 0.1 that means internal resistance is very small.
Lumped heat capacity model gives the relation of temperature with function of Biot number and Fourier
number.
… … … … (25)
Where, T = Temperature of body at any instant t-second after cooling or heating process is
started.
Ti = Initial temperature of body
T∞ = ambient temperature
XI. Froude Number
It used in calculation of fluid flow.
It is defined as the ratio of Inertial force to Gravitational force.
It is encountered for open channel flow system e.g. river , ocean.
Froude number used in calculation of hydraulic pump, wind engineering and field which related to fluid
mechanics and hydraulic activity.
Froude number , Fr = Inertial force / Gravitational force
… … … (26)
Fr = 1 → Critical flow = both waves and water flow has same velocity
Fr <1 → Subcritical flow = waves moves faster then water flow, slow and stable flow
Fr >1 → Supercritical flow = waves move slower than water flow

XII. Grashof Number
ss
It is the ratio of buoyant to viscous force acting on a fluid in the velocity boundary layer.
It’s play major role in the concept of natural convection.
Grashof number , Gr = Buoyant force / Viscous Force
… … … (27)
Where, L = is for vertical flat plate
For gases , β = 1/T
For liquid β can be calculated if variation of density with temperature at constant pressure is known.
Natural convection is used if motion and mixing caused by density variations resulting from temperature
difference within the fluid.
Due to increase in temperature density is decrease and it causes the fluid to rise. This motion caused by
buoyant force.
For a vertical flat palte, flow turns turbulent for value of Gr * Pr > 109
.
Grashof number in Mass transfer
- In case of mass transfer , natural convection is caused by concentration gradient rather than concentration
gradient.
… … … (28)
Where, β’ = (-1/ρ) * (dP/ dCa)T,P
Ca,s = concentration of species a at surface
Ca,a = concentration of species a at ambient temperature
Ca = concentration of species a
Physical Significance of Grashof Number
- It is to check the fluid flow behaviour around a new prototype to ensure its use in practical field.
- It is represent how dominant is the buoyancy force which is responsible for convection comparing to
viscous force.

Dimensionless number in chemical engineering

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