Heat loss (Dynamic systems)

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PiChart86

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Hi everybody! First time poster so hopefully this is going to be done correctly:) Unfortunately I have a very bad professor so there is zero help...
I would say I have very basic understanding of calculus, at least I have solved some first order ones. However I seem to struggle sometimes with what is actually meant in the expressions.

So for the task:

Question: How does the heat loss per unit time of a spherical body depend on it's size (radius)?
Some physics:
The heat energy of a body is given by E=kVT, where k is a (material dependent ) constant, V is body volume, and T is its temperature (in Kelvin).

A body loses heat energy according to:
dE/dt=cA(T-To)

where c is some (surface material dependent) constant, A is the body surface area, and To is the temperature of the surroundings.

Tip: Use the expression above to derive an expression for dT/dt, then plug the formulas for the area and volume of a sphere, respectively.


I don't know how to make an expression as the tip is suggesting. The expression already looks like a differential equation to me. The main problem is to get an expression with regards to dT/dt relating to dE/dt=cA(T-To).

Any help is very much appreciated.

/Christian
 
PiChart86 said:
Hi everybody! First time poster so hopefully this is going to be done correctly:) Unfortunately I have a very bad professor so there is zero help...
I would say I have very basic understanding of calculus, at least I have solved some first order ones. However I seem to struggle sometimes with what is actually meant in the expressions.

So for the task:

Question: How does the heat loss per unit time of a spherical body depend on it's size (radius)?
Some physics:
The heat energy of a body is given by E=kVT, where k is a (material dependent ) constant, V is body volume, and T is its temperature (in Kelvin).

A body loses heat energy according to:
dE/dt=cA(T-To)

where c is some (surface material dependent) constant, A is the body surface area, and To is the temperature of the surroundings.

Tip: Use the expression above to derive an expression for dT/dt, then plug the formulas for the area and volume of a sphere, respectively.


I don't know how to make an expression as the tip is suggesting. The expression already looks like a differential equation to me. The main problem is to get an expression with regards to dT/dt relating to dE/dt=cA(T-To).

Any help is very much appreciated.

/Christian
Click to expand...
Since k, V, c, A, and T0 are constant, you have two equations for dE/dt. That is, take the derivative of the first
E = k V T.
Then combine that with the expression of the second equation
dE/dt=cA(T-To)
to get a differential equation for T, i.e.
dT/dt = \(\displaystyle \alpha\) [T - T0]
What is \(\displaystyle \alpha\)? What is \(\displaystyle \alpha\) when the object is a sphere?
 
Hi and thanks for your reply!

Ok so I'm just taking a shot into the dark.

E=kVT -> dE/dT=kV -> dE=kV x dT

I plugg the expression into the second equation dE/dt=cA(T-To) -> (kV x dT)/dt=cA(T-To) -> dT/dt=cA(T-To)/kV

So your alpha is c/kV x A (formula for sphere)? Then just solving the differential equation? Did I do it right?

/Christian
 
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