Thermal & Momentum Boundary Layer
Thermal and Momentum Boundary Layer
The Boundary Layer theory was first proposed by L.Prandtl in 1904. Boundary layer theory determines the aerodynamic drag (FD) and aerodynamic lift by analyzing hydrodynamic boundary layer formation in flying vehicles. Boundary layer theory has great applications in heat transfer enhancement, i.e., thermal boundary layer formation analysis.
The study of hydrodynamic and thermal boundary layer formations over the solid surface shows the relationship between fluid flow’s frictional resistance and heat transfer characteristics, also the study of thermal and momentum diffusivity facilitates understanding of the relationship between frictional resistance of the fluid and heat transfer. In the discussion of boundary conditions in fluid dynamics, the fundamental concept lies in the interaction of velocity and temperature layers in the flow region near the boundaries. Fluid systems exhibit unique boundary condition attributes, and designers must have an understanding of these interactions to properly design a fluid system and analyze its characteristics in laminar or turbulent conditions.
What is a Boundary Layer in Boundary Layer Theory?
The boundary layer in boundary layer theory is a thin layer of fluid formed over the solid surface or internal flow through pipes due to velocity or temperature gradient. Boundary layer theory is also used to study the internal flow of fluids. The boundary layer is basically classified as :
- Hydrodynamic Boundary Layer
- Thermal Boundary Layer
Hydrodynamic Boundary Layer : Hydrodynamic boundary layer also known as momentum boundary layer or velocity boundary layer is the region defined by the velocity gradient where the flow velocity is distributed among the different fluid layers.
Hydrodynamic Boundary Layer for External Flow :
When the real fluid flows over the solid body, the fluid is adhered to the solid boundary and assumes no-slip condition. i.e., the velocity of the fluid particle which is adhered to the boundary becomes equal to the velocity of the boundary. If the boundary is stationary, the velocity of fluid particles which are adhered to the stationary boundary becomes zero. Boundary layer theory is to determine the loss of energy for fluid flow in channels. The velocity gradients will be developed perpendicular to the solid boundary. The velocity profile for the real fluid flow over the horizontal solid surface is shown below.
Fig. Development of boundary layer over a flat plate
Where in above diagram, velocity gradient- (δu/δx) & shear stress- µ(δu/δx)
Hydrodynamic boundary layer thickness (δh) :
It is defined as the distance measured perpendicular to the boundary up to which the fluid velocity becomes 99% of free stream velocity (U∞). To get a hydrodynamic boundary layer, first draw velocity profiles at every point on the solid surface and mark the point perpendicular to the boundary where the fluid has 0.99U∞. The locus of all these points is termed as Hydrodynamic boundary layer. The formation of the boundary layer over the horizontal solid surface/flat plate is shown in the figure below.
Hydrodynamic Boundary Layer for Internal Flow :
When fluid flows through the circular/non-circular pipe, the pipe’s surface’s velocity is zero, and the maximum velocity (umax) at the pipe center. The average velocity (uavg) fluid is used to analyze fluid flow characteristics. The velocity profile for the fluid which is flowing through the pipe is shown below.
Fig. Velocity boundary layer for internal flow
Thermal Boundary Layer : The thermal boundary layer is the region of fluid flow defined by the temperature gradient formed due to the thermal energy exchange among the adjacent layers.
Thermal Boundary Layer for External Flow :
When the fluid with free stream temperature (T∞) flows over the hot solid surface with a temperature (Ts). The temperature gradients will be developed perpendicular to the solid boundary.
Thermal boundary layer thickness (δt) :
It is the distance measured perpendicular to the boundary up to which the fluid temperature becomes 99% free stream temperature (T∞). To get the thermal boundary layer, first, draw temperature profiles at every point on the solid surface and mark the point perpendicular to the boundary where the fluid has a temperature of 0.99T∞. The locus of all these points is termed the thermal boundary layers. The formation of a thermal boundary layer over the horizontal solid surface/flat plate is shown in the figure below.
Fig. Thermal boundary layer for flow over hot plate
Thermal Boundary Layer for Internal Flow :
When fluid flows through the hot circular/non-circular pipe, the temperature at the pipe’s surface temperature of the hot pipe (Ts) and the minimum temperature (Tmin) at the pipe center. The average temperature (Tavg) fluid is used to analyze the thermal characteristics of fluid flow. The temperature profile for the fluid which is flowing through the pipe is shown below.
Fig. Thermal boundary layer for internal flow
Relationship between Hydrodynamic (δh) and Thermal (δt) Boundary Layer Thicknesses :
L.Prandtl proposed the relationship betweenδhand δtafter analyzing different hydrodynamic and thermal boundary layer formations in different cases. Prandtl number is a dimensionless number, and it is defined as the ratio of momentum diffusivity(ν)and thermal diffusivity(α). The relationship between hydrodynamic and thermal boundary layer thicknesses is given below.
δh/δt=(Pr)¹/3
Here,
δh= Hydrodynamic boundary layer thickness
δt= Thermal boundary layer thickness
Pr= Prandtl number =v/α
Different cases based on Prandtl number :
Case 1 : If Pr = 1, then δh = δt
The thermal boundary layer at any 𝑥 is equals the thickness of the velocity boundary layer
Case 2 : If Pr < 1, then δt > δh
The thermal boundary layer at any 𝑥 is thicker than the thickness of the velocity boundary layer.
Case 3 : If Pr > 1, then δt < δh
The thermal boundary layer at any 𝑥 is thinner than the thickness of the velocity boundary layer.
Conclusion
In our blog we have started with what is boundary layer and how it forms, we have discussed types of boundary layer i.e hydrodynamic boundary layer, thermal boundary layer also we have drawn the profile of them for internal flow and external flow through pipe. We have obtained the relationship between thickness of thermal boundary layer and hydrodynamic boundary layer and found that they are related to each other via Prandtl number so based on that we have drawn some cases.
References
- Blasius H. Grenzschichten in Flussigkeiten and Mit Kleiner reibung. Zeitschrift fur Angewandte Mathematik und Physik. 1908;56:1–37. French.
- Aziz A. A similarity solution for laminar thermal boundary layer over a flat plate with a convective surface boundary condition. Communications in Nonlinear Science and Numerical Simulation. 2009; 14:1064–1068.
- Bataller RC. Radiation effects for the Blasius and Sakiadis flows with a convective surface boundary condition. Journal of Applied Mathematics and Computation. 2008;206:832–840.
- Cortell R. Numerical solutions of the Classical Blasius flat-plate problem. Applied Mathematics and Computation. 2005;170: 706–710. 5. Cortell R. Similarity solutions for flow and heat transfer of a quiescent fluid over a nonlinearly stretching surface. Journal of Matter Process Technology. 2008;20
Informative!
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