Heat Transfer : Unsteady State Heat Conduction & Fins

By Akhil Gupta|Updated : December 26th, 2021

                                                                                                                                 

If the temperature of a body does not vary with time, it is said to be in steady state. But if there is an abrupt change in its surface temperature, it attains an equilibrium temperature or a steady state after some period. During this period, the temperature varies with time and body is said to be in the unsteady or transient state. This phenomenon is known as Unsteady or transient heat conduction.

Lumped System Analysis 

  • Interior temperatures of some bodies remain essentially uniform at all times during a heat transfer process.
  • The temperature of such bodies is only a function of time, T = T(t).
  • The heat transfer analysis based on this idealization is called lumped system analysis.

Consider a body of the arbitrary shape of mass m, volume V, surface area A, density ρ and specific heat Cp initially at a uniform temperature Ti.  

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  • At time t = 0, the body is placed into a medium at temperature T (T >Ti) with a heat transfer coefficient h. 
  • An  energy  balance  of  the  solid  for  a  time  interval  dt  can  be  expressed as:
    heat transfer into the body  during dt =  the increase in the energy of  the body during dt 
                                    h A (T – T) dt = m Cp dT 

     With m = ρV and change of variable dT = d(T – T):

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    Integrating from t = 0 to T = Ti

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Criterion for Lumped System Analysis

  • Lumped system approximation provides a great convenience in heat transfer analysis.

        We want to establish a criterion for the applicability of the lumped system analysis.

         A characteristic length scale is defined as:    Lc= V/A  (where ,V = Volumn, A = Surface area)

A non-dimensional parameter, the Biot number, is defined: 

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  • The  Biot number is the ratio of the internal resistance  (conduction)  to the external resistance to heat convection.
  • Lumped system analysis assumes a uniform temperature distribution throughout the body, which implies that the conduction heat resistance is zero. Thus, the lumped system analysis is exact when Bi = 0.
  • It is generally accepted that the lumped system analysis is applicable if: Bi ≤ 0.1
  • Therefore, small bodies with high thermal conductivity are good candidates for lumped system analysis.  Note that assuming h to be constant and uniform is an approximation.

Fourier number:
Fourier number is given by:

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06-Unsteady-heat-conduction (10)

Heat Transfer from Extended Surface (Fin):

  • A fin is a surface that extends from an object to increase the rate of heat transfer to or from the environment by increase convection.
  • Adding a fin to an object increases the surface area and can sometimes be an economical solution to heat transfer problems.
  • Finned surfaces are commonly used in practice to enhance heat transfer. In the analysis of the fins, we consider steady operation with no heat generation in the fin.
  • We also assume that the convection heat transfer coefficient h to be constant and uniform over the entire surface of the fin.
  • The rate of heat transfer from a solid surface to atmosphere is given by Q = hA ∆ T where, h and ∆T are not controllable.
  • So, to increase the value of Q surface area should be increased. The extended surface which increases the rate of heat transfer is known as fin.

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  • General equation of 2nd order: θ = c1emx + c2e–mx
  • Heat dissipation can take place on the basis of three cases.

Case 1: Heat Dissipation from an Infinitely Long Fin (L  ∞):

  • In such a case, the temperature at the end of Fin approaches to surrounding fluid temperature ta as shown in figure. The boundary conditions are given below
  • At x=0, t=t: θ = t0 – ta = θ0
  • At x =L→ ∞: t = ta, θ = 0
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    θ = θ0 e–mx
  • Heat transfer by conduction at base:

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Case 2: Heat Dissipation from a Fin Insulated at the End Tip:

  • Practically, the heat loss from the long and thin film tip is negligible, thus the end of the tip can be, considered as insulated.
  • At x=0, t=t0 and θ = t0 – ta = θ0

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Fin Efficiency:

Fin efficiency is given by:
05-Heat-transfer-through-fins_files (14)

Note: The following must be noted for a proper fin selection:  

  • The longer the fin, the larger the heat transfer area and thus the higher the rate of heat transfer from the fin.
  • The larger the fin, the bigger the mass, the higher the price, and larger the fluid friction.
  • The fin efficiency decreases with increasing fin length because of the decrease in fin temperature with length.

Fin Effectiveness: 

  • The performance of fins is judged on the basis of the enhancement in heat transfer relative to the no‐fin case, and expressed in terms of the fin effectiveness: 

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To increase fins effectiveness, one can conclude:  

  • The thermal conductivity of the fin material must be as high as possible     
  • The ratio of perimeter to the cross‐sectional area p/Ac should be as high as possible 
  • The use of fin is most effective in applications that involve low convection heat transfer coefficient, i.e. natural convection.   

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