Heat xfer -- theoretical question 1

[nbucska]

there are two vertical planes x inches apart. The planes
are infinitesimally thin and the space is filled with air. How does the heat conduction/area depends on the distance ?

<nbucska@pcperipherals.com>

 

[saxon]

nbuska, This is fundamental Thermodynamics problem in composite walls and heat transfer. The solution method is based on an electrical analogy of multiple resistances. Such that the heat flow encounters various resistances to its flow, wall1=R1 Airgap=R2, wall2=R3, so the total resistance to heat flow becomes Rt=R1+R2+R3. Now, if you increase the Airgap by increasing dim. X, the resistance R2 will also increase. By decreasing the gap, R2 will decrease.

Also, if the areas of transfer are changing, this compounds and complexes the issue slightly, but I won't go into that. Without going through all the manipulations to get to this point, the resultant formula ends up as:

Q=(Ta-Tb)/Rt

where:
Q=heat flow
Ta=initial temp.
Tb=final temp.
Rt=total of resistances to heat flow

Hope this helps.
saxon

 

[nbucska]

Sorry SAXON, let me refrase it:

How does the hear transfer across a given air layer -- due
to convection and conduction -- changes with the thickness
of the layer ?


<nbucska@pcperipherals.com>

 

[Montemayor]

nbucska:

Saxon stated the answer exactly as it is expounded in theory, except that as you stated, there is no &quot;wall resistance&quot; because you established that the planes are &quot;infinitesimally thin&quot;. This is all in keeping with the famous Fourier rate equation that electrical engineers love so dearly.

Now that you have thrown another ingredient into the pot in the shape of covection, you've really &quot;stirred up&quot; the mixture so to speak. Convection effects are unpredictable if left on their own - you can have natural convection or forced convection. To the degree that you can define it, you will establish a relationship with the amount of heat transfer due to that effect. As the air layer is widened (or becomes thicker, the effect of a fixed convection effect becomes less as does the conduction effect - although not at the same rate. The total effect is very complex and requires specific identification in order to quantify the resultant relative results.

Art Montemayor
Spring, TX

 

[prex]

This problem is treated by McAdams and certainly many other textbooks on heat transmission (unfortunately not the online book at

http://web.mit.edu/lienhard/

http://www/ahtt.html

), and you might want to consult one of them, as the problem is quite complex.
The correlation reported by McAdams depends on many factors:
-the ratio L/x, where L is the height of the enclosure and x the gap (or layer) thickness: tests were conducted for a limited range of this ratio
-the Grashof number (that in turn depends on x³)
-for N_Gr<2x10³ there is no convection, and heat transmission is due to conduction only (not accounted for by the equation)
-for 2.1x10³<N_Gr<2x10⁴ the overall heat transfer coefficient between the two walls is inversely proportional to the fourth root of x
-for 2.1x10⁵<N_Gr<1.1x10⁷ the overall heat transfer coefficient between the two walls is independent on x if the ratio L/x is kept constant
-radiation may be of course of relevance and must be accounted for separately.

This problem was of importance in calculating the heat evacuated through the double walled penetrations of sodium cooled reactors, and I know that in France they calculated also the effect of the space width (the circumference of the penetration) and of the nonuniform wall temperatures by a finite volume approach, as convection loops were found to be forming around the circumference.

prex

http://www.xcalcs.com

Online tools for structural design

 

[chicopee]

primarily by conduction if flowing air is in the laminar region otherwise the heat transfer will be by convection.

 

[poetix99]

The reply from &quot;prex&quot; will best put you on the right track. As you seemed to have known from your second question, the relationship btwn conduction and convection will change with the relative and absolute geometry.

You should consult a graduate level heat transfer text. In the case of natural convection (buoyant flows) it is critical that you regard the limits of any correlations AND the orientation with respect to gravity for such correlations.

 

[mirchee]

I recommend consulting the authoritative book &quot;Process Heat Transfer&quot; by G.F. Hewitt, G.L. Shires, and T.R. Bott (CRC Press, 1994). This is considered to be the definitive update to Kern's classic text of the same name. Natural convection correlations are discussed in Section 2.4.0 &quot;Heat Transfer Coefficients for Natural Convection&quot;, page 111-119. This book also provides equations for the effect of the aspect ratio (vertical length of plates divided by distance between plates) on convection heat transfer, leading to an important criterion for the optimal spacing for double glazed windows.

 

[corus]

Heat transfer across the air filled gap is not only by conduction/convection but also by radiation. At higher temperatures this can be significant.

 

[mirchee]

The original question discussed conduction only, and others have pointed out the need to consider convection also. As pointed out by corus, if the difference in plate temperatures is great, radiant heat transfer will also become important.

The Hewitt et. al. reference I cited also discusses radiant heat transfer in such cases in considerable detail.

Additional outstanding radiant heat transfer refences, useful for complex geometries, are:
(1) McAdams, &quot;Heat Transmission&quot; (McGraw-Hill, 3rd Edition, 1954)
(2) Hottel and Sarofim, &quot;Radiative Transfer&quot; (McGraw-Hill, 1967)
(3) Modest, &quot;Radiative Heat Transfer&quot; (McGraw-Hill, 1993)

 

[Montemayor]

To all who have stressed radiation heat transfer:

nbucska rephrased his original question by asking: &quot;How does the hear transfer across a given air layer -- due to convection and conduction -- changes with the thickness of the layer ?&quot;

I think the question is pretty specific. It limits the response to consider solely convection and conduction - no radiation is to be considered (whether it is important or not). As the title infers, this is theoretical heat transfer, not real life.


Art Montemayor
Spring, TX