One Dimensional NonHomogeneous Conduction Equation P M V

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One Dimensional Non-Homogeneous Conduction Equation P M V Subbarao Associate Professor Mechanical Engineering Department

One Dimensional Non-Homogeneous Conduction Equation P M V Subbarao Associate Professor Mechanical Engineering Department IIT Delhi Another simple Mathematical modification…. . But finds innumerable number of Applications….

Further Mathematical Analysis : Homogeneous ODE • How to obtain a non-homogeneous ODE for

Further Mathematical Analysis : Homogeneous ODE • How to obtain a non-homogeneous ODE for one dimensional Steady State Heat Conduction problems? • Blending of Convection or radiation effects into Conduction model. • Generation of Thermal Energy in a solid body. • GARDNER-MURRAY Ideas.

Blending of Convection or Radiation in Conduction Equation Continuous Convection or Radiation heat transfer

Blending of Convection or Radiation in Conduction Equation Continuous Convection or Radiation heat transfer to/from fin surface Extended surface Conduction heat transfer to /from body. Body to gain or loose heat Conduction through the fin is strengthened or weakened by continuous convection or radiation from/to fin surface.

Mathematical Ideas are More Natural An optimum body size is essential for the ability

Mathematical Ideas are More Natural An optimum body size is essential for the ability to regulate body temperature by blood-borne heat exchange. For animals in air, this optimum size is a little over 5 kg. For animals living in water, the optimum size is much larger, on the order of 100 kg or so. This may explain why large reptiles today are largely aquatic and terrestrial reptiles are smaller.

Mathematical Ideas are More Natural • Reptiles like high steady body temperatures just as

Mathematical Ideas are More Natural • Reptiles like high steady body temperatures just as mammals and birds. • They have sophisticated ways to manage flows of heat between their bodies and the environment. • One common way they do this is to use blood flow within the body to facilitate heat uptake and retard heat loss. • Blood flow is not effective as a medium of heat transfer everywhere in the body. • Body shape also enters into the equation. • It also helps expalin the odd appendages like crests and sails that decorated extinct reptiles like Stegosaurus or mammal-like reptiles like Dimetrodon. • Theoretical Biologists did Calculations to show these structures could act as very effective heat exchange fins. • These fins are allowing animals with crests to heat their bodies up to high temperatures much faster than animals without them.

Amalgamation of Conduction and Convection/Radiation Heat Convection In/out Heat Conduciton in Heat Conduciton out

Amalgamation of Conduction and Convection/Radiation Heat Convection In/out Heat Conduciton in Heat Conduciton out

Basic Geometric Features of Fins profile PROFILE AREA cross-section CROSS-SECTION AREA

Basic Geometric Features of Fins profile PROFILE AREA cross-section CROSS-SECTION AREA

Innovative Fin Designs

Innovative Fin Designs

Single Fins : Shapes Longitudinal or strip Radial Pins

Single Fins : Shapes Longitudinal or strip Radial Pins

Anatomy of A STRIP FIN Di rec tio n Dx Flo w thickness x

Anatomy of A STRIP FIN Di rec tio n Dx Flo w thickness x

GARDNER-MURRAY ANALYSIS : ASSUMPTIONS w w w w w Steady state one dimensional conduction

GARDNER-MURRAY ANALYSIS : ASSUMPTIONS w w w w w Steady state one dimensional conduction Model. No Heat sources or sinks within the fin. Thermal conductivity constant and uniform in all directions. Heat transfer coefficient constant and uniform over fin faces. Surrounding temperature constant and uniform. Base temperature constant and uniform over fin base. Fin width much smaller than fin height. No bond resistance between fin base and prime surface. Heat flow off fin proportional to temperature excess.

Slender Fins Dx thickness x

Slender Fins Dx thickness x

Steady One-dimensional Conduction through Fins qconv or qradiation qx Conservation of Energy: OR qx+dx

Steady One-dimensional Conduction through Fins qconv or qradiation qx Conservation of Energy: OR qx+dx

Where OR

Where OR

Substituting and dividing by Dx: Taking limit Dx tends to zero and using the

Substituting and dividing by Dx: Taking limit Dx tends to zero and using the definition of derivative: Substitute Fourier’s Law of Conduction:

Fins with Cartesian Geometry Heat Transfer L x qb Straight fin of triangular profile

Fins with Cartesian Geometry Heat Transfer L x qb Straight fin of triangular profile rectangular C. S. b x=b x qb x=0 Straight fin of parabolic profile rectangular C. S. L b x=0

Fins with Cartesian Geometry Heat Transfer Straight fin of triangular profile Circular C. S.

Fins with Cartesian Geometry Heat Transfer Straight fin of triangular profile Circular C. S. Straight fin of parabolic profile rectangular C. S.

Fins with Cylindrical Geometry Heat Transfer Circumferential fin of rectangular profile Straight fin of

Fins with Cylindrical Geometry Heat Transfer Circumferential fin of rectangular profile Straight fin of triangular profile

For a constant cross section area:

For a constant cross section area:

Fin factor for pin Fin: Fin factor for strip Fin:

Fin factor for pin Fin: Fin factor for strip Fin:

Define: At the base of the fin:

Define: At the base of the fin:

Tip of A Fin

Tip of A Fin

Linear Second order ODE with Constant Coefficients • This equation has two linearly independent

Linear Second order ODE with Constant Coefficients • This equation has two linearly independent solutions. • The general solution is the linear combination of those two independent solutions. • Each solution function qi(x) and its second derivative must be constant multiple of each other. • Therefore, the general solution function of the differential equation above is:

At the base of the fin: Infinitely long fin: Logic from Mathematics shows that

At the base of the fin: Infinitely long fin: Logic from Mathematics shows that C 1 = 0 !

At the base of the fin: For a strip fin:

At the base of the fin: For a strip fin:

Rate of Heat Transfer in an Infinitely Long Strip Fin

Rate of Heat Transfer in an Infinitely Long Strip Fin

Most Practicable Boundary Condition b b Dx Corrected adiabatic tip: thickness x

Most Practicable Boundary Condition b b Dx Corrected adiabatic tip: thickness x

Longitudinal Fin : Adiabatic Tip The boundary condition are: Using these gives: and The

Longitudinal Fin : Adiabatic Tip The boundary condition are: Using these gives: and The foregoing shows that:

With the general solution for the temperature excess And from the previous slide

With the general solution for the temperature excess And from the previous slide

The heat flow through the fin at any location x is: And at x=b

The heat flow through the fin at any location x is: And at x=b (heat entering fin base): For a strip fin:

Efficiency of Strip Fin The fin efficiency, h, is defined as the ratio of

Efficiency of Strip Fin The fin efficiency, h, is defined as the ratio of the actual heat dissipation to the ideal heat dissipation if the entire fin were to operate at the base temperature excess

For infinitely long strip fin: For Adiabatic strip fin:

For infinitely long strip fin: For Adiabatic strip fin:

Strip Fin: Infinitely Long

Strip Fin: Infinitely Long

Strip Fin: Adiabatic tip

Strip Fin: Adiabatic tip

SUMMARY Longitudinal Fin of Rectangular Profile: adiabatic tip w Temperature Excess Profile w Heat

SUMMARY Longitudinal Fin of Rectangular Profile: adiabatic tip w Temperature Excess Profile w Heat Dissipated = Heat Entering Base w Fin Efficiency