Double Pipe HEAT EXCHANGERS with Low Thermal Resistance P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Ideas for Creation of Isotropically Compact HX!!!

Need for Compact HXs Double Pipe Hxs are long, even for moderate capacities. Unviable to accommodate in an industrial space. The size of heat exchanger is very large in those applications where gas is a medium of heat exchange. Continuous research is focused on development of Compact Heat Exchangers --- High rates of heat transfer per unit volume. The rate of heat exchange is proportional to –The value of Overall heat transfer coefficient. –The surface area of heat transfer available. –The mean temperature difference.

Large surface area Heat Exchangers The use of extended surfaces will reduce the gas side thermal resistance. To reduce size and weight of heat exchangers, many compact heat exchangers with various fin patterns were developed to reduce the air side thermal resistance. Fins on the outside the tube may be categorized as –1) flat or continuous (plain, wavy or interrupted) external fins on arrays of tubes, – 2) Normal fins on individual tubes, –3) Longitudinal fins on individual tubes.

Innovative Designs for Extended Surfaces

Geometrical Classification Longitudinal or strip RadialPins

Anatomy of A STRIP FIN thickness x xx Flow Direction

profile PROFILE AREA cross-section CROSS-SECTION AREA Basic Geometric Features of Longitudinal Extended Surfaces

Complex Geometry in Nature 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.

Straight fin of triangular profile rectangular C.S. Straight fin of parabolic profile rectangular C.S. L x=0 b x=b x qbqb L b qbqb x=0 x Longitudinal Extended Surfaces with Variable C.S.A

For a constant cross section area:

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

Rate of Heat Transfer through a constant Area Fin Fin Efficiency:

How to decide the height of fin for a Double Pipe HX ?

Strip Fin of Least Material The heat flux is not constant throughout the fin surface area. It decreases as some function of distance from the fin base. Two models are possible: For a constant heat flux, the cross-section of the fin must also decrease as some function of distance from the base. Schmidt reasoned that the problem reduced to the determination of a fin width function,  (x), that would yield minimum profile area.

Longitudinal Fin of Least Material Constant Heat Flux Model Consider With A a function of x. Then For a constant heat flux (with k a constant by assumption): and which is a linear temperature excess profile. The practical feasibility of this solution depends on ease of manufacturing.

Least profile area for a given rate of heat transfer can be modified as maximum rate of heat transfer for a given profile area A p For a Longitudinal fin of Rectangular Cross Section with L = 1: Strip Fin of Least Material : OPTIMUM SHAPES ( L=1) With, let Hence

Optimum Shapes : Strip Fin Find the best shape where and get Solving iteratively gives  R = Find the optimum shape for a given A p

LONGITUDINAL FIN OF CONCAVE PARABOLIC PROFILE The differential equation for temperature excess is an Euler equation: L b qbqb x=bx=a=0 x

The particular solution for temperature excess is: And the heat dissipation (L=1) is: Efficiency:

Gardner’s curves for the fin efficiency of several types of longitudinal fins. Longitudinal Fins

n th order Longitudinal Fins

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Performance of Optimum Profiles : Strip Fin(L=1) Heat dissipated Optimum fin width (mb=1.4192)

Performance of Optimum Profiles Optimum shape for a given q b &  b And solve for A p with [ tanh (1.4192) = ]

Selection of Fin Material Rectangular Profile: Consider three popular materials: Steel Aluminum Copper

Selection of Fin Material For a given length, fin mass is proportional to A p. A p is inversely proportional to thermal conductivity. For given h,  b, and q b :

Comparison of Longitudinal Fin profile area varies as the cube of To double the heat flow, you use two fins or make one fin eight times as large. There is a virtue in using short stubby fins.

LONGITUDINAL FIN OF TRIANGULAR PROFILE The differential equation for temperature excess : L x=a=0 b x=b x qbqb