In this lesson we will talk about some key concepts in Heat conduction. Let’s start
our discussion with One-dimensional steady-state solutions to the heat conduction equation.
Even though, it is a simple expression, it gives us several useful practical observations
and relations. Consider a planar wall of thickness L which is thin in the x-direction. For this
system, the heat conduction equation reduces to an ordinary differential equation as shown
here. Integrating this and using the boundary conditions to determine the constants of integration,
we get the final solution in the form of a linear distribution of the temperature. Then,
using Fourier’s law we can determine the heat flux as shown here. Now let’s include
a constant volumetric heat source in our planar slab model. A very good example of this is
the case of an electrical resistance heater where the heat generated by the flow of electricity
through a resistance element, like a defrosting wire in a rear window of a car. The 1d conduction
equation now includes a source term as shown here. Integrating and applying boundary conditions
we obtain a temperature distribution shown here. As you can see the temperature distribution
is parabolic instead of linear and the temperature reaches its peak within the wall at the location
corresponding to this. Now lets go back to the case without a volumetric
heat source. The linear nature of the planar heat conduction, and its relation to heat
flow is analogous to electrical resistance in a wire under a potential difference. Thus,
the conduction solution can be written in a form identical to Ohm’s law for electrical
current, where, in the equivalent thermal circuit the heat flow corresponds to the current
flow, temperature drop corresponds to voltage drop and thermal resistance corresponds to
electrical resistance. This analogy is a very powerful tool as it allows us to easily formulate
and compute the total thermal resistance and heat transfer rates across complex composite
walls which can be converted to thermal resistance networks. One thing to keep in mind is that,
eventhough, we can extend the serial thermal resistance methodology to serial-parallel
wall structures, the results obtained from such an analysis will only be approximate,
as the heat conduction in most cases will be two dimensional or three-dimensional in
nature. A similar 1-d approach can be used for the
analysis of heat conduction in solid cylindrical tubes, but using the cylindrical-polar coordinates.
Applications of this analysis are flows of liquids, through pipes of a heating system
or a heat exchanger. Consider a heated solid tube as shown here. The steady-state conduction
equation in cylindrical polar coordinate reduces to a 1 dimensional ordinary differential equation
in terms of the radial coordinates which can be integrated to obtain the following temperature
distribution. Again, using the Fourier’s law we can get the following expression for
the radial heat transfer for a tube of length L. As with the planar conduction, the heat
flow is constant with the thermal resistance given by this expression. Using a similar
approach as in the case of planar conduction, we can analyze heat transfer through a composite
tube comprising of multiple concentric layers of material as shown here.
Another important class of problems involve heat transfer via a uniform convective boundary
condition imposed on the boundary of a slender rod or plate. This application is referred
to as an extended surface or fin. Fins have many engineering applications including dissipating
heat from heat exchangers used in HVAC, engine cooling, electrical systems and electronics
packaging. In fact, there is evidence suggesting that even dinosaurs used fins for thermoregulatory
functions. From Newton’s law of cooling, we can model the heat loss from a surface
as a local heat flux. Here we are assuming that the temperature of the fin at position
x is uniform over its cross-section, so that the surface temperature equals the local fin
temperature. This is needed to apply our 1-D modeling and is valid for relatively thin
fins. Also, the convective heat transfer coefficient h is assumed to be known constant value, but
that assumption can be relaxed if the variation in h is known. We can now apply a more general
quasi-one-dimensional approach when fin’s cross-sectional area Ac can vary with the
longitudinal distance as shown here. Keep in mind that this variation is small enough
to assume a slender ‘slice’ of the fin as a 1D element. Applying the conservation
of energy to this slice, we can say that the heat conduction into the element at x is equal
to the rate of heat conduction from the element at x+dx and rate of heat convection from the
slice. Using the Fourier’s law and the newton’s law of cooling we get the following general
form of heat transfer equation for a quasi-1D extended surface. In general, the cross-sectional
area Ac and the perimeter P of the fin vary with x making this differential equation difficult
to solve. However, in the special case of constant cross-section such as rectangular
plates or circular pins and constant thermal conductivity the differential equation reduces
to this. This is a linear, homogenous, second-order differential equation with constant co-efficient.
The general solution of this equation is shown here, where the constants C1 and C2 are determined
using the boundary conditions. The boundary conditions at the fins tip can be prescribed
in several ways leading to different forms of the solution as shown in this table. The
key takeaway here is that depending on the prescribed boundary condition at the tip,
we get different temperature distributions and heat transfer rate from the fins. Lets
consider a long pin fin of a heat sink having a diameter of 3 mm. The fin base temperature
is 80 degree Celsius whereas the ambient temperature is 20 degrees Celsius, with a heat transfer
coefficient of 50 watts per meter square. Assuming the fin is long enough for its end
to attain ambient temperature of 20 degrees Celsius, the temperature distribution is given
by the following expression. Using this we can plot the axial variation of temperature
for fins made of different materials. As Copper has a highest thermal conductivity among the
three, we can see that the copper fin has the highest overall temperature compared to
the other two. As a result, it will allow the highest heat transfer to the ambience
and out of the heat sink. Note that temperature distribution asymptotically tends to the ambient
temperature, and extending fin length has diminishing effect on heat transfer.
So far in our discussion we assumed steady heat conduction, but this assumption might
not be valid for a wide range of heat transfer problems that are intrinsically unsteady for
eg cooling of metal molds during the solidifying process, or the heating of a cold car on a
winter morning, cooling of a cup of hot coffee. The
main goals of conduction heat transfer in such applications is to determine temperature
distribution inside the object as a function of space and time, and to predict the overall
time it will take the object to attain a certain temperature during the heating or cooling
process. Let’s start our analysis by examining the unsteady conduction equation for constant
properties. Its solution will now be a function of both space and time, and properties such
as density, specific heat, and thermal conductivity will influence the time scales of unsteadiness
i. e. how fast or slow a given transient occurs. Consider a small body being cooled by convection
from a cold fluid. If we can assume that the temperature distribution inside the body is
nearly uniform then the temperature gradients in the body are very small and we can neglect
the spatial derivative term. The release of energy by the solid can be modeled as volumetric
heat sink as shown here. This transient conduction equation can be written using the temperature
difference like this, which has a simple exponential solution. Using the initial condition, we
can get the final solution as this equation which shows the exponential decay type nature
of this problem. Lets consider a stainless steel sphere of 0.1 m diameter in a turbulent
airflow of 10 m/s. Using these material properties we can compute beta, and this would give us
the following variation of temperature over time. We can see that for this case, the lumped
model predicts that it will take about 2 hours to cool the sphere to within 1percent of the
ambient temperature. Now, you may ask, when can we use this type
of simplified analysis? The answer to this question lies in a dimensionless number called
Biot Number. Consider a solid object being baked in an oven. As the solid is being heated
by the hotter fluid surrounding it, the heat is first convected to the body and subsequently
conducted within the body. The Biot number is the ratio of the internal resistance of
a body to heat conduction to its external resistance to heat convection. Therefore,
a small Biot number represents small resistance to heat conduction and thus small temperature
gradients within the body. The Lumped system analysis assumes a uniform temperature distribution
throughout the body, which can only happen when the thermal resistance of the body to
heat conduction is zero. Thus, theoretically, lumped system analysis is exact only when
Bi=0 and approximate when Bi>0. Of course, it goes without saying, the smaller the Biot
number, the closer it is to zero, the more accurate the lumped system analysis will be.
It is generally accepted that the lumped system analysis is applicable if Bi is less than
or equal to 0.1. The exponent of the non-dimensional lumped
model equation can be re-written as this, which after some rearrangement can be written
as the product of two dimensionless numbers, the Biot number and the Fourier number. The
fourier number is a non-dimensional time parameter which, together with the Biot number is used
to characterize unsteady conduction problems. To understand the physical significance of
the Fourier number, lets rearrange the expression a bit. As you can see, the Fourier number
is a measure of heat conducted through a body relative to the heat storage. Thus, a large
value of the Fourier number indicates faster propagation of heat through a body. Before
we wrap up this lesson, lets look at the electrical analogy of the transient thermal response.
For an electrical circuit with a capacitor and a resistor, the transient voltage is given
by this expression. If we compare this with the lumped thermal solution, we see that if
we introduce a characteristic length scale L, then beta can be interpreted as this expression.
Therefore, we see that Re is the equivalent thermal resistance and Ce is the equivalent
thermal capacitance. With that, we have come to the end of this lesson.