Heat Transfer of a Cable and Air

[Crater]

I'm in need of an equation that relates the loss of heat (in power) of a cable to ambient air to various parameters such as diameter, temperatures of air and cable, and all other variable that may play a role.

My research online has produced some results, but I haven't quite found what I'm looking for. I figured I'd run it by here and see if someone had the correct equation on hand.

Thanks for any help.

 

[xez]

I may have some useful equations on hand which I could
look up this evening, but it occurs to me that there are a
couple of unspecified factors that are potentially HUGELY
relevant to the answer.

You say 'cable'... well some cables, e.g. 'wire' are made of
solid metal with a smooth round exterior, and have no
insulation / jacketing.

Some 'cables' are made of woven / twisted braids of wire,
and hence they have a very different volume to surface
area parameter, and their surface is quite non-smooth.

Some 'cables' have jacketing, insulating coating, rubberized
or plastic coating, et. al.

Some 'cables' are assembiles of many individual 'wires' or
subsidiary elements all encompassed in various forms of
jacketing / shielding materials.

Of course some 'cables' in the physical mechanics sense are
made of things like rope, plastic, glass, or whatever and
aren't principally metal / wire at all.

Until you clarify the general material composition, coating,
length / diameter, etc. of the cable, it'll be hard to
correctly define the appropriate answer. Since you say
there's a heat / power loss, I assume you mean it's
an electrical cable, though certainly stressed or heated
non-electrical cables can have power / heat dissipation
needs too.

There should be ample references about the temperature
rise of various forms of bare or lightly insulated wire/cable
in electrical safety / transmission engineering references.

There's a strong dependence in heat dissipation of course
depending on whether the cable is vertical, horizontal,
hanging in free space, laying on a surface, coiled, confined
in a small conduit / duct, whether there's a certain amount
of airflow other than that due to convection currents due
to the heating itself, etc.

Of course the cable resistivity and thermal conductivity
plays a role so it's good to know if it's copper, aluminium,
etc.

 

[berkeman]

Welcome to the PF, Crater. Here is a recent thread where we discussed a similar question in a lot of detail. I don't know if it will help you out or not, but maybe it will lead you to some useful ways to treat this problem:

https://www.physicsforums.com/showthread.php?t=168438

 

[Crater]

Thank you for you responses.

Summary of Below: I need an equation for the power handling of a microwave coaxial cable instead of the thermal one I spoke of earlier.



I am sorry for not clarifying earlier, as I myself was assigned to look for the vague equation I mentioned. By clarifying exactly what I now know what to look for, I will essentially be changing the information I requested.

The cable in question is a microwave coaxial transmission cable. I am looking to find an equation for the power handling capability of said cable. I know this to be first determined by a maximum voltage (before arcing occurs) and maximum average voltage. The average voltage is dictated by the thermal extremes with which the cable can withstand before damage occurs. Orginally, I sought a means to determine how quickly heat would be released to the environment as a partial and round-about way to derive part of the power handling equation. This was pretty foolish. Since it is much more likely that there is a fully derived equation for the power handling of a microwave coaxial cable, I've adjusted my search to that.


Thank you again for the help you have provided. I'm currently at my first internship with only a year of college completed so far. The help of others is greatly appreciated.

 

[berkeman]

The coax cable should have a maximum power rating in its detailed datasheet. Do you have the datasheet for the cable?

 

[Crater]

We actually design and produce coaxial cables.

I'm updating our in house power handling calculator (Excel based), as it is currently producing innaccurate results. The current equation used for it doesn't seem to synch up with other partial equations I've found, nor depend on enough variables. Between other work I've been assigned I've been looking through a lot of free information on the net and stuff in our archives to find the equation I'm looking for. The perfect article always seems to be for sale only.

 

[xez]

Well my ITT reference data for radio engineers 4th ed.
says that:

Voltage gradient in a coaxial line
At the voltage standing wave maximum:
(gradient at surface of inner conductor) =

5.37/d * sqrt( SP_kw / (Z_0 * epsilon) )

d = diameter of inner conductor

SP_kw = power in killowatts at the crest of the modulation
cycle, thus if the carrier is 1kW and modulation 100% set
SP_kw = 4kW.

Z_0 = characteristic impedance of the coaxial line in ohms.

epsilon = effective relative permittivity of the coaxial line;
air = 1.0.

sqrt(x) = x^0.5 = square root.

The same page cites the breakdown strength of air
at atmospheric pressure as 29000 peak Volts/cm
(experimental value before derating).

The same source also cites the equation:
delta_E / delta_r =
0.434*E / (r * log_10(D/d)
== which also ==
0.059*E*C / (r * epsilon)
== which also ==
60*E / (r * Z_0 * sqrt(epsilon) )

C = capacitance between coaxial line conductors in
picofarads per foot

D = diameter of inner surface of outer conductor in
same units as d.

d = diameter of inner conductor.

E = total voltage across line (E and delta_E both
RMS or both peak)

r = radius from cable center to inner surface of outer
conductor (r and delta_r both in same units)

Now my personal comments ----

You'll have to derate the dielectric breakdown strength
to be much less as either temperature increases or as
frequency increases. There are some empirical data
points to suggest how a certain material's RF breakdown
strength may be derated vs. temperature and frequency,
but in general you'll never find the data for just exactly
the frequency and more importantly the material blend /
characteristics you have, so this is likely something to
be empirically measured at failure and then very
conservatively derated for safety / variance margins.

The actual steady state heat dissipation of the cable will
depend in a complex way on its construction and
dielectric core and jacketing materials. It's probably
easier just to use experiments of both DC and RF power
through a reasonable length (several feet?) of the cable
and use an IR temperature measurement camera and/or
small thermocouple probes on the cable's surface to
measure the temperature rise of the cable versus time and
to determine the steady state "free still air" equilibrium
temperature value that is reached. Of course the interior
dielectric temperature would be much higher, and perhaps
with a DC test you could insert a tiny thermocouple into
the middle of the cable to get a sense of the
temperature gradient between the cable's jacket and the
cable's core. Doing a resistance change measurement
across the cable's core wire may help deduce the
core wire temperatue too if the measurement can be
accurate enough.

Depending on the ambient temperature of environmental
operation during the test, the cable materials,
and the frequencies and powers involved you'll either
reach a point where parts of the cable (e.g. plastics)
start to melt, or a point where there's RF parameter
degradation/instability and ultimately an RF arcing type
of fault in the cable. Low frequencies will tend to arc less
readily than higher ones according to conventional wisdom.
It's perhaps possible to have the cable be fairly
"melty" inside even before you get a detected arc failure,
though as long as the dielectric keeps insulating the cable
even in a semi-molten state.

So the end story is that you'll have to
determine/specify certain thermal conductivity,
thermal gradient, and thermal rise wrt. ambient temperature parameters to help determine
what temperatur related factors are limitations relative
to your cables materials softening, melting,
catching fire, etc. That's about the same DC as RF.

Relevant to RF, you'll have to determine over the
frequencies and powers of interest at what point the
dielectic strength and RF impedance of the cable may
start to degrade such that it no longer can safely or
effectively transport the signal without arcing or deforming.
You should be able to test the thing pretty easily with
a high power VHF / UHF source like a magnetron or
powrer oscillator tube and a variable power coupling
arrangement or something like that, and maybe use a
circulator or digital sampling o-scope or something to look
for reflection / distortion type events that'd indicate arcing
or degradation of the cable's RF characteristic.

You could use a simple 2D heat transfer FEM model
of the cross section of the cable containing regions for
core wire, dielectic, shield, jacketing to get some idea of
the heat transport parameters of your cable but I wouldn't
be surprised if those were inaccurate by 2:1 or 3:1
depending on the fine details of your materials heat
conductivities and smoothness and so on. Measuring
the heat dissipation and conductivity with an
DC experimental setup will give you empirical data that's
a lot better in many ways than what a generic
equation / model will provide. Then you can fit a curve
or polynomial to the empirical data to generate an
equation that may more accurately extrapolate / interpolate
relative to other similar cable variants you may have.

With respect to equations, models, standards,
test specifications, or tables of empirical
data for other coaxial cables, check MIL-SPECs,
ITU, NEC, and 1915 - 1970 era publications from places
like Bell Telephone Laboratories, MIT, Bell System
Technical Journal / Technical reports, and so on.

The basic problem is that usually things are specified
*very* conservatively, so the end number you generate
as a power handling rating will be just some semi-arbitrary
fraction of the empirically determined numbers at which
the cable goes into thermal runaway (temperature
increases without limit due to insufficient free air
convective dissipation), melts or readily arcs at RF.

e.g. You could specify an free air environmental
operating temperature range of -20 to +70C and
say that 100C is the limiting specified operational
temperature for the interior of the cable, so that'd give
room for a 30C rise from free air to cable core, then you
could see what kinds of powers generate such a rise
in the cable. Maybe your dielectic softens considerably
at 130C, so that'd be your structural oriented safety
margin assuming that the RF dielectic strength didn't
seriously degrade at the maximum operating frequency
much before the mechanical softening point of the
dielectic...

Good luck; it's good to see more / better attention being
paid to specifiying the characteristics of products; a lot
of the time I'd swear that they're just making up the
numbers and that there's no chance in heck of a product
actually *working* under its rated operating conditions.

You might also ask around over at
http://www.microwaves101.com
there are probably some old timers who could rattle
off test specifications / standards, empirical figures,
and approximation equations to you from memory.

 

[Crater]

Thank you xez for all the input.

The method we currently employ to determine power handling does take many of these things into consideration. Particularly derating factors, which currently rely on VSWR, altitude and temp. Upon closer inspection of precedents in Excel, it seems one the main problems may just be a network of estimations resulting in a fairly innaccurate number. This is mostly due to all factors being rolled up into 3 variables which don't seem to correctly relate all the parameters determining the power handling value.

However, this equation hasn't been put into use yet for determining power handling estimations. The current power handling data we supply to customers is much more in line with MIL-Specs and I believe to be based on labratory data. I do not think I will be permitted to do the recommended indepth thermal testing, though I am now interested in attempting it. I'm thinking a talk with the Test Engineer to discuss the current handling graphs we have catalouged here may produce a better equation, without taking too much work.

I will keep looking into all the sources you have mentioned in search of equations I can punch in, compare with MIL-Spec results, and hopefully find a fairly accurate one. I'll also be sure to ask around at Microwaves101. I found that website my second day on the job, and it has been quite helpful.

Thanks again!