Draft:Scale analysis of external forced convection

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Scale Analysis of External Forced Convection

Introduction

External forced convection refers to the heat transfer process where a fluid flows over a surface due to external forces, such as fans or wind, and convection occurs as a result of this forced motion. In engineering, it is important to optimize heat transfer characteristics, and this is often achieved through boundary layer theory and scale analysis, which simplify the complex phenomena of fluid flow and heat transfer.

Boundary Layer and Flow Assumptions

The boundary layer is a thin region near the surface of a solid body where viscosity and thermal conduction effects dominate. Outside this region, the flow is largely unaffected by the surface and is called the free stream. The development of the boundary layer is characterized by the Reynolds number (Re), defined as:

$Re_{x}={\frac {U_{\infty }x}{\nu }}$

where:

• $U_{\infty }$ is the free-stream velocity,
• $x$ is the distance from the leading edge of the plate,
• $\nu$ is the kinematic viscosity of the fluid.

The boundary layer thickness $\delta$ and thermal boundary layer thickness $\delta _{T}$ are the characteristic lengths that describe the extent of velocity and temperature changes within the boundary layer.

Governing Equations

The key equations governing forced convection are the continuity, momentum (Navier - Stokes), and energy equations. For an incompressible, constant-property fluid, these equations are simplified for boundary layer analysis.

The continuity equation in two dimensions is:

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

The momentum equation in the $x$-direction (longitudinal direction) is:

$u{\frac {\partial u}{\partial x}}+v{\frac {\partial u}{\partial y}}=-{\frac {1}{\rho }}{\frac {\partial P}{\partial x}}+\nu {\frac {\partial ^{2}u}{\partial y^{2}}}$

The energy equation for thermal conduction and convection is:

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

where:

• $u$ and $v$ are the velocity components in the $x$ and $y$ directions,
• $P$ is the pressure,
• $\nu$ is the kinematic viscosity,
• $\alpha$ is the thermal diffusivity, and
• $T$ is the temperature.

Scale Analysis of Momentum Equation

Scale analysis involves estimating the order of magnitude of each term in the governing equations. For the momentum equation, let:

• $u\sim U_{\infty }$ (the free-stream velocity),
• $x\sim L$ (the length of the plate),
• $y\sim \delta$ (the boundary layer thickness).

In the boundary layer, the convective inertia terms scale as:

${\frac {U_{\infty }^{2}}{L}},{v}\cdot {\frac {U_{\infty }}{\delta }}$

The viscous term, dominant in the boundary layer, scales as:

$\nu {\frac {U_{\infty }}{\delta ^{2}}}$

Equating the orders of magnitude of the convective and viscous terms leads to the following relation for the boundary layer thickness:

$\delta \sim {\frac {L}{\sqrt {Re_{x}}}}$

This shows that the boundary layer thickness grows with $x$ and decreases with the square root of the Reynolds number.

Thermal Boundary Layer

The thermal boundary layer develops similarly to the velocity boundary layer but depends on the Prandtl number (Pr), which is the ratio of momentum diffusivity to thermal diffusivity:

$Pr={\frac {\nu }{\alpha }}$

For $Pr=1$, the velocity and thermal boundary layers have the same thickness. However for $Pr\neq 1$, the thicknesses differ.

Case 1: Prandtl Number ($Pr\gg 1$)

For fluids with high Prandtl numbers, such as oils (e.g., 50–2000), the thermal boundary layer is much thinner than the velocity boundary layer. In this case:

$\delta _{T}\ll \delta$

This occurs because thermal diffusivity is much smaller than momentum diffusivity, causing heat to penetrate only a small distance into the fluid. The relationship between the boundary layers is:

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

This inverse relationship shows that for large Prandtl numbers, momentum effects dominate over thermal effects.

The heat transfer coefficient $h$ can be estimated using the thickness of the thermal boundary layer:

$h\sim {\frac {k}{\delta _{T}}}\sim {\frac {k}{L}}Re_{L}^{1/2}Pr^{1/3}$

where $k$ is the thermal conductivity of the fluid. The dimensionless Nusselt number (Nu), which is the ratio of convective to conductive heat transfer, is:

$Nu_{x}={\frac {hx}{k}}\sim Re_{L}^{1/2}Pr^{1/3}$

Case 2: Prandtl Number ($Pr\ll 1$)

For low Prandtl number fluids, such as liquid metals (e.g., 0.001–0.03), the thermal boundary layer is much thicker than the velocity boundary layer. In this case:

$\delta _{T}\gg \delta$

The large thermal diffusivity allows heat to diffuse far into the fluid. The relationship between the boundary layers is:

$\delta _{T}\sim \delta Pr^{-1/2}$

This inverse relationship shows that for small Prandtl numbers, thermal effects dominate over momentum effects.

The heat transfer coefficient $h$ can be estimated using the thickness of the thermal boundary layer:

$h\sim {\frac {k}{\delta _{T}}}\sim {\frac {k}{L}}Re_{L}^{1/2}Pr^{1/2}$

where $k$ is the thermal conductivity of the fluid. The dimensionless Nusselt number (Nu), which is the ratio of convective to conductive heat transfer, is:

$Nu_{x}={\frac {hx}{k}}\sim Re_{L}^{1/2}Pr^{1/2}$

The above scaling solutions agree within a factor of order unity with the classical analytical results (similarity solution or exact solution).

Conclusion

Scale analysis provides a simplified yet accurate approach to understanding forced convection in boundary layers. It helps estimate critical parameters such as boundary layer thickness, heat transfer coefficient, and skin friction, which are essential for applications like heat exchangers and aerodynamic surfaces. By examining the relationship between the hydrodynamic and thermal boundary layers for different Prandtl numbers, we can design systems that optimize heat transfer and minimize drag.

References

• Incropera, F.P., DeWitt, D.P., Bergman, T.L., & Lavine, A.S. (2007). *Introduction to Heat Transfer* (5th ed.). John Wiley & Sons. ISBN 978-0-471-45728-2.
• Schlichting, H., & Gersten, K. (2017). *Boundary-Layer Theory* (9th ed.). Springer. ISBN 978-3-662-52918-7.
• Bejan, A. (1995). *Convection Heat Transfer* (2nd ed.). Wiley-Interscience. ISBN 978-0-471-50292-0.
• White, F. M. (2006). *Viscous Fluid Flow* (3rd ed.). McGraw-Hill. ISBN 978-0-07-240231-4.
• Cengel, Y.A., & Ghajar, A.J. (2014). *Heat and Mass Transfer: Fundamentals and Applications* (5th ed.). McGraw-Hill. ISBN 978-0-07-339818-0.
• Kays, W.M., & Crawford, M.E. (1993). *Convective Heat and Mass Transfer* (3rd ed.). McGraw-Hill. ISBN 978-0-07-034703-7.
• Rohsenow, W.M., Hartnett, J.P., & Cho, Y.I. (1998). *Handbook of Heat Transfer* (3rd ed.). McGraw-Hill. ISBN 978-0-07-053555-0.
• Eckert, E.R.G., & Drake, R.M. (1972). *Analysis of Heat and Mass Transfer*. McGraw-Hill. ISBN 978-0-07-019756-7.
• Blasius, H. (1908). "Boundary Layer Theory." *ZAMM - Journal of Applied Mathematics and Mechanics*, 1(1): 1–7. DOI: 10.1002/zamm.19080010102.

Article prepared by

This part of the article on Scale Analysis of External Forced Convection was jointly developed by the following students from IIT BHU (Varanasi):

• Abhishek Gupta (Roll No. 21135002)
• Amit Sharma (Roll No. 21135016)
• A C Jashwanth Rao (Roll No. 21135022)
• Arin Dhaulakhandi (Roll No. 21135026)