re: insteady-state heat transfer... SOLAR.KnowledgePublications.com

re: insteady-state heat transfer...
30 apr 1997
tino kirschner <100663.1015@compuserve.com> wrote:

>hello!

greetings tino,

>i have a problem with the heat transition of a gearbox. i want to compare
>the thermal properties of a gearbox whith a steel cast housing and a alu
>(die) cast housing. imagine the gearbox get a constant input energy and
>had a constant output. if the gearbox starts to run the temperature inside
>(oil-temperature) increase till a point is reached where the temerature has
>a steady condition. i want to compare this behaviour between the two kinds
>of housing. i'm sure that the behavour is different, cause the heat
>transmission coefficient of alu is higher than steel.

about 4 times higher, but will the al be thicker, since it isn't so strong?
assuming the al housing conducts heat better... 
 
>1. in the phase where the temperature is not in steady condition, i think,
>that the gearbox whith the alu-housing should have a higher temperature
>at the outer surface of the housing and a lower temperature inside
>(oil-temperatur) as the same gearbox whith the steel-cast housing.
>is this right?

i think so, but maybe not much, if the outside is air-cooled, and the air is
the dominant series thermal resistance for gearbox heat dissipation. still air
thermal resistance is about 2000 times greater than steel. al has a lower heat
capacity by volume than steel, so the housing would heat up slightly faster. 

>2. the gearbox whith the alu-housing should reach the point of steady
>temperature condition much earlier! is this right?

not much earlier. that depends on the thermal mass of the oil and gears as
well. steel is about like water, by volume, about 60 btu/f-ft^3, vs about 40
for al and about 25 for oil. if the gearbox, oil and gears all have equal
heat capacitance, the gearbox itself will only contribute 1/3 to the time
constant. say it's a 20 hp gearbox, 99% efficient, with 180 btu/f of heat
capacitance and 5 ft^2 of air-cooled gearbox surface. the heat input is about
500 btu/hr and still air has a thermal conductance of about 1.5 btu/h-f-ft^2,
so the steady-state temp would be about 70 + 500/(5x1.5) = 137 f. the rc time
constant with steel is 180/(5x1.5) = 24 hours, so its temperature changes as
t(t) = 137 - (137-70)exp(-t/24), where t is in hours, never quite reaching
137, but reaching 130 f when t = -24ln((137-130)/(137-70)) = 54 hours.

changing the gearbox to al makes c=160 btu/f, rc=21 hours, and t=48 hours.

>3. if we have a steady temperature condition, the temperature of the
>surface of the housing should be the same (alu and steel). is this right?

close to the same, i think, if the housing shapes are the same. 

>4. if we have a steady temperature condition, the temperature inside the
>housing (oil-sump) should be the same (alu and steel). is this right?

close to the same, i think. 
 
>5. how can i calculate this behaviour?

use "ohm's law for heatflow," including conductive paths, to find the
steady state temperature, along with that exponential equation. 

nick

nicholson l. pine                      system design and consulting
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computer simulation and modeling. high performance, low cost, solar heating and
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