HYDRODYNAMIC  BOUNDARY

LAYER QUESTION ANSWERS 

It is a 3-dimensional space

over the flat plate. There is a

velocity variation in it. There is

a zero velocity t the plate surface.

It has free velocity at the boundary.

Q. Define velocity and thermal boundary layers. Draw relevant sketches.

Fig. Velocity Boundary Layer

The Velocity boundary is the 3-dimensional space over the flat plate in which velocity is variable. It is zero at the solid surface and is free flow velocity at the upper boundary of the BL.

CONSTRUCTION

It is a 3 dimensional space between a surface (Flat plate/pipe) and the boundary in which temperature is varying. There is a temperature gradient in this space. Normally there is a high temperature at the solid surface and low temperature fluid is moving over it. It causes a temperature gradient. Flow can be laminar or turbulent. The boundary layer thickness vary all along the length of the solid surface. It is δ_th1  at distance x1 & δ_th2 at x2, and so on.

FEATURES

There is no heat transfer outside the boundary layer. Boundary layer is thin. Boundary layer shows the variation in temperature in convention.. It depends on velocity. It varies parabolically. Therefore temperature also in the boundary layer varies in a parabolic way.

Q.  Which parameter governs the thickness of the velocity boundary layer ?

Reynolds number

Q. Define thermal boundary layer thickness.

In forced convective heat transfer, there is a temperature gradient between the hot surface and the fluid flowing over it. The distance from the plate to the free flowing fluid is thermal boundary layer thickness. Its symbol is δ_th. Its units will be mm. It is a function of distance ‘x’ from the leading edge. It is also related to velocity boundary thickness δ and Pr number.

δ_th = δ x Pr^–1/3

Q. Give the relation between thickness of velocity boundary layer and thermal boundary layer.

 For laminar flow, the relation is

δ_th =δ_ve Pr^–1/3

Q. Explain why the thermal boundary layer is much more than the velocity boundary layer in liquid metals.

 Liquid metals have very low Prandtl number because of their high thermal conductivity.

Q. When is the thermal boundary layer much smaller than the velocity boundary layer?

 When Prandtl number is high i.e. for an oil.

 Q. What is thermometric well and draw its sketch?

The pocket in a pipe welded radially. Insert a thermo-couple/thermometer to measure temperature of a fluid flowing inside the pipe. It is a thermometric well.

Q. How error appears while measuring temperature with thermometric well?

Say fluid temperature inside the pipe is higher than the ambient temperature; heat transfer starts from the fluid to the thermometric well. Then heat flows  from the bottom of the thermometric well to the wall of the pipe. Then it goes to the atmosphere. Hence temperature at the bottom of the well is not the true temperature of the fluid. That is how error comes in.

Q. How to minimize error in temperature measurement in a thermometric well?

Consider thermometric well as a fin with tip insulated. Equation for temperature distribution in such a case is

(T_x –T_f)/ (T_base —T_f) =Cosh [m (l-x)]/Cosh (ml)

At x=l We get

(T_l –T_f)/ (T_base —T_f) =    1/Cosh (ml)

Now T_l is the temperature at the bottom of the well

Perimeter of the well P = (d+ 2δ) A_cs = Pi.x dδ P/A_cs

                                        = 1/δ m = (hP/kA_cs)^0.5

Substituting, we get

m = (h/kδ)^0.5

Therefore error minimizes by increasing h and decreasing k and δ. Achieve it by using long and thin well. Use a liquid in the thermometer pocket.

 

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