1. AFE BABALOLA UNIVERSITY
HEAT AND MASS TRANSFER By
Engr. Saheed Akande
11/26/2017
AFE BABALOLA UNIVERSITY

2. One-Dimensional Steady State Conduction with Internal Heat Generation
Energy generation per unit volume q = E/V
1. The plane wall
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PLANE WALL
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Cylinder
Heat transfer across a cylinder
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Applications:
There are a number of situations in which there are sources of heat in the domain of interest. Examples are:
Electrical heaters where electrical energy is converted resistively into heat
current carrying conductors
chemically reacting systems
nuclear reactors
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WORKED EXAMPLE
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solution
Assumptions: (1) steady-state conditions, (2) one –dimensional heat flow, (3) constant properties.
Analysis: (a) the appropriate form of heat equation for steady state, one dimensional condition with constant properties is
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Solution (contd)
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Solution (contd)
(b) The heat fluxes at the wall faces can be evaluated from Fourier’s law,
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Using the temperature distribution T(x) to evaluate the gradient, find
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Comments: from an overall energy balance on the wall, it follows that
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ATTEMPT
The steady state temperatures distribution in a one-dimensional wall of thermal conductivity and thickness L is of the form T=ax3+bx2+cx+d. derive expressions for the heat generation rate per unit volume in the wall and heat fluxes at the two wall faces(x=0, L).
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13. Cylinder with heat source
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14. Extended Surface Heat Transfer
Heat transfer between a solid surface and a moving fluid (convection) is governed by the Newton’s cooling law: q = hA(Ts-T∞), whereTs is the surface temperature and T∞ is the fluid temperature. Thus, to increase the convective heat transfer, one can
i. Increase the temperature difference (Ts-T∞) between the surface and the fluid.
ii. Increase the convection coefficient h. This can be accomplished by increasing the fluid flow over the surface since h is a function of the flow velocity and the higher the velocity, the higher the h. Example: a cooling fan.
iii. Increase the contact surface area A. Example: a heat sink with fins.
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Many times, when the first option is not in our control and the second option (i.e. increasing h) is already stretched to its limit, we are left with the only alternative of increasing the effective surface area by using fins or extended surfaces. Fins are protrusions from the base surface into the cooling fluid, so that the extra surface of the protrusions is also in contact with the fluid.
Common examples encountered cooling fins includes
Air-cooled engines (motorcycles, portable generators, etc.),
electronic equipment (CPUs), automobile radiators, air
conditioning equipment (condensers) and elsewhere.
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16. Heat Transfer From a Fin
Fins are used in a large number of applications to increase the heat transfer from surfaces. Typically, the fin material has a high thermal conductivity. The fin is exposed to a flowing fluid, which cools or heats it, with the high thermal conductivity allowing increased heat being conducted from the wall through the fin. The design of cooling fins is encountered in many situations and we thus examine heat transfer in a fin as a way of defining some criteria for design.
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University of Ibadan, Ibadan, Nigeria

17. Extended surface analysis
Assumption : steady state One-Dimensional analysis
Extended surface of interest
rectangular or pin fins of constant cross sectional area.
Annular fins or fins involving a tapered cross section area : This may involve solution of more complicated equations. Numerical methods of integration or computer programs can be used to advantage in such cases.
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Consider the cooling fin shown below:
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24. Solution of the Fin Equation
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Other fin condittions
We have discussed, the long fin and the insulated tip. Two other possibilities are usually considered for fin analysis:
(i) a tip subjected to convective heat transfer, and
(ii) a tip with a prescribed temperature.
The expressions for temperature distribution and fin heat transfer for all the four cases
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FIN EFFECTIVENESS
How effective a fin can enhance heat transfer is characterized by the fin effectiveness, f ε , which is as the ratio of fin heat transfer and the heat transfer without the fin. For an adiabatic fin:
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If the fin is long enough, mL>2, tanh(mL)→1, and hence it can be considered as infinite fin (case D in Table 3.1). Hence, for long fins,
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Fin Efficiency
The fin efficiency is defined as the ratio of the energy transferred through a real fin to that transferred through an ideal fin.
An ideal fin is one made of a perfect or infinite conductor material.
A perfect conductor has an infinite thermal conductivity so that the entire fin is at the base material temperature.
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44. Overall fin efficiency
Overall fin efficiency for an array of fins
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46. Composite slab (Contd)
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dissimilar materials (Parallel thermal resistance)
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Heat transfer across an insulated wall (parallel thermal resistance)
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51. One Dimensional steady state radial conduction
The steady state (quasi) one-dimensional equation that has been developed ( for planar) can also be applied to radial (non-planar) geometries. An important case often encountered in situations where fluids are pumped and heat is transferred that would be explored are
Cylindrical surfaces
Spherical surfaces
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52. One Dimensional steady state radial conduction
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55. ONE DIMENSIONAL STEADY STATE HEAT CONDUCTION
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EXPRESSIONS FOR ONE DIMENSIONAL STEADY STATE HEAT CONDUCTIONHEAT TRANSFER
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60. APPLICATION OF ONE DIMENSIONAL STEADY STATE HEAT CONDUCTION
Heat transfer through a plane slab
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University of Ibadan, Ibadan, Nigeria
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MODES OF HEAT TRANSFER
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65. Thermal Resistance Circuits (Electrical analogy)
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Compare equation 13 to ohms law
V = IR
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