**Content: Prandtl Number**

**Prandtl Number**

**Prandtl Number formula**

The Prandtl (Pr) number formula is given by

[latex]Pr=\frac{Momentum \;diffusivity}{Thermal \;Diffusivity}[/latex]

[latex]\\Pr=\frac{\mu C_p}{k}\\ Pr=\frac{\nu}{\alpha}[/latex]

Where:

μ = dynamic viscosity

C_{p} = Specific Heat of the fluid taken into consideration

k = Thermal Conductivity of the fluid

ν = Kinematic viscosity

α = Thermal diffusivity

ρ = Density of the fluid

Prandtl (Pr) number is independent of Length. It depends upon the property, Type and state of the fluid. It gives the relation between the viscosity and thermal conductivity.

Fluids having Prandtl (Pr) number in the Lower spectrum are free-flowing fluids and generally possess high thermal conductivity. They are excellent as heat conducting liquids in heat exchanger and similar applications. Liquid metals are brilliant in heat transfer. As viscosity increases Prandtl (Pr) number increases and thus heat conduction capacity of fluid decreases.

**Physical significance of Prandtl Number**

During heat-transfer in-between wall and a flowing-fluid, heat is transferred from a high-temperature wall to the flowing fluid thru a momentum boundary-layer that comprises the bulk-fluid substance and a transitional and a thermal boundary-layer that comprises of stationary film. In the stagnant film, heat transfer occurs by conduction in the fluid. The importance of Prandtl (Pr) number of the flowing fluid is to taken into account as it relates momentum boundary-layer to the thermal one during heat-transfer through the fluid.

when Prandtl (Pr) number has Small values, Pr << 1, It represents that thermal diffusivity dominating over momentum diffusivity and liquid metal has lower Prandtl (Pr) number and heat diffuses significantly faster in that. Thermal boundary layer has higher thickness comparation of velocity-based boundary-layer in liquid-metal.

Similarly, for large values of Prandtl (Pr) number, Pr >> 1, the momentum diffusivity dominates over thermal diffusivity. oils have higher Prandtl (Pr) number and heat diffuses slowly in oils. Thermal boundary layer has Lower thickness relative to velocity boundary layer in oils.

For liquid mercury the heat conduction is more dominant in comparison to convection, Thus thermal diffusivity is dominant in Mercury. Though, for engine-oil, convection is highly effective in heat transfer from a high temperature area when compared to purely conduction case, thus, momentum diffusivity is significant parameter in Engine-oil.

Gases lie in the middle of this spectrum. Their Prandtl (Pr) number is about 1. Thermal boundary layer has equal thickness relative to velocity boundary layer.

The ratio of the thermal to momentum boundary layer over a flat plate is given by the following equation

[latex]\frac{\delta_t}{\delta}=Pr^\frac{-1}{3}\;\;\;\;\;\;\;0.6<Pr<50[/latex]

**Magnetic Prandtl Number**

Magnetic Prandtl Number is a dimensionless number which gives the relation between Momentum diffusivity and magnetic diffusivity. It is the ratio of viscous diffusion rate to the magnetic diffusion rate. It generally occurs in magnetohydrodynamics. It can also be evaluated as the ratio of magnetic Reynold’s Number to the Reynold’s Numbers.

[latex]\\Pr_m=\frac{\nu}{\eta} \\Pr_m=\frac{Re_m}{Re}[/latex]

Where,

Rem is the magnetic Reynolds number

Re is the Reynolds number

ν is the viscous diffusion rate

η is the magnetic diffusion rate

**Prandtl Number Heat Transfer**

when Prandtl (Pr) number has Small values, Pr << 1, It represents that thermal diffusivity dominating over momentum diffusivity. Liquid metal has lower Prandtl (Pr) number and heat disseminates very quickly in Liquid metal and Thermal-boundary layer is much thicker in comparison to velocity-boundary layer in liquid-metal.

Similarly, for large values of Prandtl (Pr) number, Pr >> 1, the momentum diffusivity dominates over thermal diffusivity. oils have higher Prandtl (Pr) number and heat diffuses slowly in oils. Thermal boundary layer has Lower thickness relative to velocity boundary layer in oils.

For liquid mercury the heat conduction is more dominant in comparison to convection, Thus thermal diffusivity is dominant in Mercury. Though, for engine’s oil, convection is highly effective in heat-transfer from a high temperature area when compared to purely conductive, thus, momentum diffusivity is significant in Engine’s oil.

Gases lie in the middle of this spectrum. Their Prandtl (Pr) number is about 1. Thermal boundary layer has equal thickness relative to velocity boundary layer.

The ratio of the thermal to momentum boundary layer over a flat plate is given by the equation

[latex]\frac{\delta_t}{\delta}=Pr^\frac{-1}{3}\;\;\;\;\;\;\;0.6<Pr<50[/latex]

**Turbulent Prandtl Number**

The turbulent Prandtl number Pr_{t} is a dimensionless term. It is the ratio of momentum eddy diffusivity to the heat transfer eddy diffusivity and utilized for the evaluation of heat transfer for turbulent boundary layer flow condition.

**Does heat transfer coefficient depend on Prandtl number?**

Heat Transfer coefficient is also calculated by means of Nusselt’s Number. This is represented by the ratio of Convective heat transfer to the conductive heat transfer.

For forced convection,

[latex]Nu=\frac{hL_c}{k}[/latex]

Where,

h = the convective heat transfer coefficient

L_{c} = the characteristic length,

k = the thermal conductivity of the fluid.

Also, Nusselt Number is the function of Reynold’s Number and Prandtl (Pr) number. Thus, Change in Prandtl (Pr) number changes the Nusselt Number and thus heat transfer coefficient.

**Does Prandtl number change with pressure?**

Prandtl (Pr) number is assumed to be independent of pressure. Prandtl (Pr) number is a function of Temperature since μ,C_{p} are the function of Temperature but a very weak function of pressure.

**Effect of Prandtl number on boundary layer | ****Effect of Prandtl number** on heat transfer

**Effect of Prandtl number**on heat transfer

when Prandtl (Pr) number has Small values, Pr << 1, It represents that thermal diffusivity dominating over momentum diffusivity. Liquid metals have lower Prandtl (Pr) number and heat diffuses very quickly in Liquid metals. Thermal boundary layer has higher thickness relative to velocity boundary layer in Liquid metals.

Similarly, for large values of Prandtl (Pr) number, Pr >> 1, the momentum diffusivity dominates over thermal diffusivity. oils have higher Prandtl (Pr) number and heat diffuses slowly in oils. Thermal boundary layer has Lower thickness relative to velocity boundary layer in oils.

For liquid mercury the heat conduction is more dominant in comparison to convection, Thus thermal diffusivity is dominant in Mercury.

Gases lie in the middle of this spectrum. Their Prandtl (Pr) number is about 1. Thermal boundary layer has equal thickness relative to velocity boundary layer.

**Prandtl number of Air**

Prandtl (Pr) number for Air is given below in the table

Prandtl (Pr) number of Air at 1 atm pressure, temperature °C is given as:

Temperature | Pr |

[°C] | Dimensionless |

-100 | 0.734 |

-50 | 0.720 |

0 | 0.711 |

25 | 0.707 |

50 | 0.705 |

100 | 0.701 |

150 | 0.699 |

200 | 0.698 |

250 | 0.699 |

300 | 0.702 |

**Prandtl number of Water at different Temperatures**

Prandtl (Pr) number of Water in Liquid and vapor form at 1 atm Pressure is shown below:

Temperature | Pr number |

[°C] | Dimensionless |

0 | 13.6 |

5 | 11.2 |

10 | 9.46 |

20 | 6.99 |

25 | 6.13 |

30 | 5.43 |

50 | 3.56 |

75 | 2.39 |

100 | 1.76 |

100 | 1.03 |

125 | 0.996 |

150 | 0.978 |

175 | 0.965 |

200 | 0.958 |

250 | 0.947 |

300 | 0.939 |

350 | 0.932 |

400 | 0.926 |

500 | 0.916 |

**Prandtl number of Ethylene glycol**

Prandtl (Pr) number of Ethylene glycol is Pr = 40.36.

**Prandtl number of Oil | Prandtl number of Engine Oil**

Prandtl (Pr) number for oil lies between the range of 50-100,000

Prandtl (Pr) number of Engine Oil at 1 atm Pressure are given below:

**Prandtl number Table**

Temperature (K) | Pr number |

260 | 144500 |

280 | 27200 |

300 | 6450 |

320 | 1990 |

340 | 795 |

360 | 395 |

380 | 230 |

400 | 155 |

**Prandtl number of Hydrogen**

Prandtl (Pr) number of Hydrogen at 1 atm Pressure and at 300 K is 0.701

**Prandtl number of Gases | Prandtl Number of Argon, Krypton etc.**

**Prandtl number of Liquid Metals and other Liquids**

**Benzene Prandtl number**

Prandtl (Pr) number of Benzene at 300 K is 7.79.

**CO**_{2} Prandtl number

_{2}Prandtl number

Prandtl (Pr) number of Hydrogen at 1 atm Pressure is 0.75

**Prandtl number of Ethane**

Prandtl (Pr) number of Ethane is 4.60 in Liquid form and 4.05 in gaseous form

**Gasoline Prandtl number**

Prandtl (Pr) number of Gasoline is 4.3

**Glycerin Prandtl number**

Prandtl (Pr) number of Glycerin lies between the range of 2000-100,000

**Some Important FAQs**

**Q.1 How is Prandtl number calculated?**

Ans: Pr Number can be calculated by using the formula

[latex]Pr=\frac{\mu C_p}{k}[/latex]

Where:

- μ = dynamic viscosity
- C
_{p}= Specific Heat of the fluid taken into consideration - k = Thermal Conductivity of the fluid

**Q.2 What is the value of Prandtl number for liquid metals?**

Ans: The Prandtl (Pr) number for Liquid metals is extremely Low. Pr<<<1. For example In liquid mercury has Prandtl (Pr) number = 0.03 which represents that, the heat conduction is more dominant in comparison to convection, Thus thermal diffusivity is dominant in Mercury.

**Q.3 What is the Prandtl number of Water?**

Ans: Prandtl (Pr) number of Water in Liquid and vapor form at 1 atm Pressure is shown below:

Temperature | Prandtl (Pr) number |

[°C] | Dimensionless |

0 | 13.6 |

5 | 11.2 |

10 | 9.46 |

20 | 6.99 |

25 | 6.13 |

30 | 5.43 |

50 | 3.56 |

75 | 2.39 |

100 | 1.76 |

100 | 1.03 |

125 | 0.996 |

150 | 0.978 |

175 | 0.965 |

200 | 0.958 |

250 | 0.947 |

300 | 0.939 |

350 | 0.932 |

400 | 0.926 |

500 | 0.916 |

**Q.4 What does Prandtl number represent?**

Ans: During the heat transfer amongst a wall-barrier and fluid, heat is transferred from a high-temp barrier to fluid through a momentum-boundary-layer. This includes fluids and a transitional and a thermal boundary-layer that comprises of film. In the stagnant film, heat transfer happens by fluid’s conduction on that time. The * Pr* number of the flowing fluid, is ratio which taken into account of momentum boundary layer to the thermal boundary layer.

**Q.5 what is the Prandtl Number for Steam?**

Ans: The Prandtl (Pr) number for steam at 500 C is 0.916.

**Q.6 what is the Prandtl Number for Helium?**

Ans: Prandtl (Pr) number of Helium is 0.71

**Q.7 what is the Prandtl Number for Oxygen?**

Ans: Prandtl (Pr) number of Oxygen is 0.70

**Q.8 What is the Prandtl Number for Sodium?**

Ans: Prandtl (Pr) number of Sodium is 0.01

**Q.9 How is the Prandtl number related with kinematic viscosity and thermal diffusivity?**

Ans: The **Prandtl (Pr) number** is well-defined as the ratio of momentum diffusivity to thermal diffusivity.

Its formula is given by:

The Pr Number formula is given by

[latex]Pr=\frac{Momentum \;diffusivity}{Thermal \;Diffusivity}[/latex]

[latex]\\Pr=\frac{\mu C_p}{k}\\ Pr=\frac{\nu}{\alpha}[/latex]

Where:

μ = dynamic viscosity

C_{p} = Specific Heat of the fluid taken into consideration

k = Thermal Conductivity of the fluid

ν = Kinematic viscosity

[latex]\nu=\frac{\mu}{\rho}[/latex]

α = Thermal diffusivity

[latex]\alpha=\frac{k}{\rho C_p}[/latex]

ρ = Density of the fluid

From the above formula we can say that Prandtl (Pr) Number is inversely proportional to Thermal diffusivity and directly proportional to Kinematic viscosity.

**Q.10 Is there any fluid which has a Prandtl number in the range of 10 20 except water?**

Ans: There are certain number of fluids that has Prandtl (Pr) Number in the range of 10-20. They are Listed below:

- Acetic acid [Pr = 14.5] at 15C and [Pr = 10.5] at 100C
- Water [Pr = 13.6] at 0C
- n-Butyl Alcohol is [Pr = 11.5] at 100 C
- Ethanol [Pr = 15.5] at 15C and [Pr = 10.1] at 100C
- Nitro Benzene [Pr = 19.5] at 15C
- Sulfuric acid at high concentration about 98% [Pr = 15] at 100C

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