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Transport Phenomena

Difference Between Laminar Boundary Layers for Low and High Prandtl Number Fluids Using Scale Analysis

The formation of hydrodynamic and thermal boundary layers is crucial in understanding fluid dynamics and heat transfer near solid surfaces. Here's a detailed explanation of both layers:

Hydrodynamic Boundary Layer

1. Definition: The hydrodynamic boundary layer is a region of fluid flow where the effects of viscosity are significant. It is characterized by a velocity gradient from zero at the solid surface (due to the no-slip condition) to the free-stream velocity away from the surface.

2. Formation Process:

• Flow Development: When a fluid flows over a solid surface, the fluid in immediate contact with the surface experiences friction, leading to a velocity of zero at the surface.
• Velocity Gradient: Moving away from the surface, the velocity increases due to the inertia of the incoming fluid. This creates a gradient of velocity within the boundary layer.
• Boundary Layer Thickness: The thickness of the hydrodynamic boundary layer increases with distance from the leading edge of the surface. It is influenced by factors such as fluid viscosity, flow velocity, and surface roughness.

Thermal Boundary Layer

1. Definition: The thermal boundary layer is a region adjacent to a solid surface where the temperature of the fluid changes from the surface temperature to the free-stream temperature.

2. Formation Process:

• Heat Transfer: When heat is transferred between a surface and the fluid, a temperature gradient forms. The fluid in contact with the surface gains or loses heat, affecting its temperature.
• Temperature Gradient: The thermal boundary layer develops as the temperature of the fluid transitions from the surface temperature to that of the bulk fluid.
• Heat Transfer Mechanisms: Both conduction (heat diffusion through the fluid) and convection (bulk movement of the fluid) contribute to the temperature distribution in the thermal boundary layer.

3. Boundary Layer Thickness: The thickness of the thermal boundary layer is influenced by the rate of heat transfer, fluid properties (like thermal conductivity and specific heat), and flow conditions. It may differ from the hydrodynamic boundary layer, especially in turbulent flows.

In summary, the hydrodynamic boundary layer forms due to viscous effects that create a velocity gradient in the fluid, while the thermal boundary layer develops due to heat transfer between the fluid and the surface, resulting in a temperature gradient.

Types of Flow

• Laminar Flow: In laminar conditions, the flow is smooth and orderly, leading to a well-defined boundary layer with a predictable velocity profile.
• Turbulent Flow: In turbulent conditions, the flow is chaotic, resulting in a thicker boundary layer and more complex velocity profiles.

We will be dealing with only laminar flow.

Convection

Convection is the process of thermal energy exchange in fluids through the motion of matter within them. It occurs in both gases and liquids and leads to a cyclical effect.

Types of Convection

• Natural Convection: Fluid moves due to natural means, such as the buoyancy effect. This occurs when warmer fluid rises and cooler fluid falls due to temperature differences in the fluid.
• Forced Convection: Fluid is forced to move over a surface or in a tube by external means, such as a pump or fan.

Prandtl Number

The Prandtl number (Pr) is a dimensionless quantity, crucial in understanding the interplay between hydrodynamic and thermal boundary layers.

It is defined as the ratio of momentum diffusivity (ν) to thermal diffusivity (α):

Pr = ν / α

Where:

• ν represents the kinematic viscosity, a measure of the fluid's resistance to shear flow.
• α represents the thermal diffusivity, a measure of the fluid's ability to conduct heat.

The Prandtl number significantly influences the relative thicknesses and development of the boundary layers.

• Low Prandtl Number (Pr < 1): When Pr < 1, thermal diffusivity exceeds momentum diffusivity, signifying that heat transport is more efficient than momentum transport. Consequently, the thermal boundary layer grows faster and thicker than the hydrodynamic boundary layer. Liquid metals, such as mercury, exhibit this behaviour. Lower Pr values, indicating higher thermal diffusivity, lead to thicker thermal boundary layers and enhanced heat transfer rates.

• High Prandtl Number (Pr > 1): When Pr > 1, momentum diffusivity dominates over thermal diffusivity; this indicates that momentum is transported more effectively than heat. As a result, the hydrodynamic boundary layer develops more rapidly and becomes thicker than the thermal boundary layer. This is typical of fluids like water, oils, and other liquids. Conversely, higher Pr values result in thinner thermal boundary layers and reduced heat transfer rates.

• Prandtl Number Equal to 1 (Pr = 1): When Pr = 1, momentum and thermal diffusivities are equal, implying that momentum and heat are transported equally efficiently. Consequently, the hydrodynamic and thermal boundary layers grow at similar rates and have comparable thicknesses. This scenario is observed in gases like air.

1. FORCED CONVECTION

FOR LOW PRANDTL NUMBER

Governing equations:

${\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}=0$ .........1

$u{\frac {\partial u}{\partial x}}+v{\frac {\partial u}{\partial y}}=\left({\frac {1}{\rho }}\right){\frac {dp}{dx}}+\nu {\frac {\partial ^{2}u}{\partial y^{2}}}$ ..........2

$u{\frac {\partial T}{\partial x}}+v{\frac {\partial T}{\partial y}}=\alpha {\frac {\partial ^{2}T}{\partial y^{2}}}$ ........................3

In eq (2), neglecting $\left({\frac {1}{\rho }}\right){\frac {dp}{dx}}$ term and performing scale analysis we get:

${\frac {U^{2}}{L}}\sim \nu {\frac {U}{\delta ^{2}}}$

${\frac {\delta ^{2}}{L^{2}}}\sim {\frac {\nu }{UL}}$

${\frac {\delta }{L}}^{2}\sim \left({\frac {UL}{\nu }}\right)^{-1}$

${\frac {\delta }{L}}\sim \left(Re_{L}\right)^{-{\frac {1}{2}}}$ ........................4

From scale analysis of equation (3):

${\frac {U\Delta T}{L}}\sim \alpha {\frac {\Delta T}{\delta _{t}^{2}}}$

${\frac {\delta _{t}^{2}}{L^{2}}}\sim {\frac {\alpha }{UL}}$

${\frac {\delta _{t}^{2}}{L^{2}}}\sim \left({\frac {\alpha }{\nu }}\right)\left({\frac {UL}{\nu }}\right)$

${\frac {\delta _{t}^{2}}{L^{2}}}\sim Pr^{-1}Re_{L}^{-1}$

${\frac {\delta _{t}}{L}}\sim Pr^{-{\frac {1}{2}}}Re_{L}^{-{\frac {1}{2}}}$ .........................5

From equations (4) and (5) we get:

${\frac {\delta _{t}}{\delta }}\sim Pr^{-{\frac {1}{2}}}$

Hence if $Pr\ll 1$, then $\delta \ll \delta _{t}$.

FOR HIGH PRANDTL NUMBER

From equation (3):

$u\Delta T/L\sim \alpha \Delta T/\delta _{t}^{2}$

We know by geometry, $u=U\delta _{t}/\delta$.

${\frac {U\delta }{L\delta }}\sim {\frac {\alpha }{\delta _{t}^{2}}}$

$\left({\frac {\delta _{t}}{L}}\right)^{3}\sim {\frac {\alpha \delta }{UL^{2}}}$

$\left({\frac {\delta _{t}}{L}}\right)^{3}\sim {\frac {\alpha }{\nu }}\cdot {\frac {\delta }{L}}\cdot {\frac {UL}{\nu }}$

$\left({\frac {\delta _{t}}{L}}\right)^{3}\sim Pr^{-1}Re_{L}^{-1/2}Re_{L}^{-1}$

$\left({\frac {\delta _{t}}{L}}\right)^{3}\sim Pr^{-1}Re_{L}^{-3/2}$

${\frac {\delta _{t}}{L}}\sim Pr^{-1/3}Re_{L}^{-1/2}\quad {\text{..................................... 6}}$

From equations 4 and 6 we get:

${\frac {\delta _{t}}{\delta }}\sim Pr^{-1/3}$

Hence if $Pr\gg 1$, then $\delta \gg \delta _{t}$.

2. NATURAL CONVECTION

Governing equations:

${\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}=0$ .................................................... 7

$u{\frac {\partial u}{\partial x}}+v{\frac {\partial u}{\partial y}}=\nu {\frac {\partial ^{2}v}{\partial x^{2}}}+g\beta (T-T_{\infty })$ .................................................... 8

In Equation 8, the LHS term is the inertia effect, the first term of the RHS is the friction effect, and the second term of the RHS is the buoyancy effect.

Order of Inertia term: $\left({\frac {H}{\delta _{t}}}\right)^{4}Ra_{H}^{-1}Pr^{-1}$

Order of friction term: $\left({\frac {H}{\delta _{t}}}\right)^{4}Ra_{H}^{-1}$

By taking the order of the buoyancy term as 1, because buoyancy is the main driving force in natural convection,

Where $Ra_{H}={\text{Rayleigh Number}}={\frac {g\beta \Delta TH^{3}}{\alpha \nu }}$.

$u{\frac {\partial T}{\partial x}}+v{\frac {\partial T}{\partial y}}=\alpha {\frac {\partial ^{2}T}{\partial x^{2}}}$ .................................................... 9

From the scaling of equation 9,

${\frac {v\Delta T}{H}}\sim {\frac {\alpha \Delta T}{\delta _{t}^{2}}}$

$v\sim {\frac {\alpha H}{\delta _{t}^{2}}}$ ..................... 10

FOR HIGH PRANDTL VALUE:

At hydrodynamic boundary layer, Friction is balanced by inertia:

Hence, by scaling of equation 8 at hydrodynamic boundary layer by neglecting buoyancy effect, we get:

${\frac {v^{2}}{H}}\sim {\frac {\nu v}{\delta ^{2}}}$
${\frac {v}{H}}\sim {\frac {\nu }{\delta ^{2}}}$

Putting the order of $v$ from equation 10, we get:

${\frac {\alpha H}{\delta _{t}^{2}H}}\sim {\frac {\nu }{\delta ^{2}}}$
${\frac {\delta _{t}}{\delta }}\sim Pr^{-1/2}$

Hence, if $Pr\gg 1$, then $\delta \gg \delta _{t}$.

FOR LOW PRANDTL VALUE NUMBER:

At thermal boundary layer, inertia effect will be balanced by buoyancy effect.

Hence, order of inertia $\sim$ order of buoyancy

$\left({\frac {H}{\delta _{t}}}\right)^{4}Ra_{H}^{-1}Pr^{-1}\sim 1$
$\delta _{t}\sim H(Ra_{H}Pr)^{-1/4}$ .............................................11

Putting this value in equation 10, we get:

$v\sim \left({\frac {\alpha }{H}}\right)(Ra_{H}Pr)^{1/2}$ .............................................12

At the hydrodynamic boundary layer, the friction effect will be balanced by buoyancy.

Hence, by scaling of equation 8:

${\frac {\nu v}{\delta _{v}^{2}}}\sim g\beta \Delta T$

Putting the value of $v$ from equation 12, we get:

$\delta _{v}\sim H(Ra_{H}/Pr)^{-1/4}$ .............................................13

From equations 11 and 13, we get:

${\frac {\delta _{t}}{\delta _{v}}}\sim Pr^{-1/2}$

Hence if $Pr\ll 1$, then $\delta \ll \delta _{t}$.

References

1. "Prandtl Number - an overview" | [ScienceDirect Topics](https://www.sciencedirect.com/topics/chemical-engineering/prandtl-number)

2. Forsberg, Charles H. (2021). "Forced convection," *Heat Transfer Principles and Applications*, Elsevier, pp. 211–266. doi:10.1016/b978-0-12-802296-2.00006-8

3. Deason, Hilary. "Science and Technology Supplements: The McGraw-Hill Yearbook of Science and Technology." Science. 153 (3737). doi:10.1126/science.153.3737.731

Article prepared by

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1. Akshat Rao, Roll no.: 21135012, IIT (BHU) Varanasi
2. Ashutosh Verma, Roll no.: 21135035, IIT (BHU) Varanasi
3. Avinash Kumar Jha, Roll no.: 21135037, IIT (BHU) Varanasi
4. Binanda Rabha, Roll no.: 21135044, IIT (BHU) Varanasi
5. Shashwat Mishra, Roll no.: 21135120, IIT (BHU) Varanasi