Unizor - Physics4Teens - Energy - Heat Transfer Problems

Notes to a video lecture on http://www.unizor.com



Heat Transfer - Problems



Problem 1



Determine the power of the heat source inside the room required to
maintain a certain difference between inside and outside air
temperature, given the following:

(a) the difference between inside and outside temperature δ=T_out−T_home

(b) the room has only one wall facing the outside air, and the area of this wall is A

(c) the thickness of the wall is L

(d) the wall is made of solid material with a coefficient of thermal conductivity k.



Solution



Let x be the distance from a point inside the wall to its surface facing the room.

The T(x) is a temperature at this point as a function from this distance.

To be equal to T_home at the surface facing the room and to be T_out at the surface facing the outside and to be linearly changing from one value to another inside the wall of the thickness L, the temperature inside the wall at distance x from the surface facing the room should be

T(x)=T_home+(T_out−T_home)·x/L

From this we can determine the heat flux through the wall at a distance x, using the Fourier's law of thermal conduction:

Q(x,A) = −k·A·dT(x)/dx =

= −k·A·(T_out−T_home)/L =

= −k·A·δ/L

That is exactly how much heat we need to maintain the difference between temperature in the room and outside.



Problem 2



Calculate the heating requirement of the room with only one concrete
wall facing outside with no windows, assuming the following:

(a) the temperature in the room must constantly be T_room=20°C

(b) the temperature outside is also constant T_out=5°C

(c) the thickness of the concrete wall facing outside is L=0.2m

(d) the area of the wall facing outside is A=12m²

(e) heat conductivity of concrete is k=0.6W/(m·°K)



Solution



Using the above, we can determine the heat flux through the wall at a distance x:

Q(x,A) = −k·A·δ/L

(for any distance x from the surface of the wall that faces the room)

Outside surface of the wall is at 5°C, inside is at 20°C.

So, δ=15°.

Therefore, the heat flux through the wall (at any distance from the inside surface) will be

Q = 0.6·12·15/0.2 = 540W

That is exactly how much heat we need to maintain the temperature in the room.



Problem 3



Our task is to determine the law of cooling of a relatively small hot
object immersed in the cool infinitely large reservoir with liquid or
gaseous substance.

We assume the volume of substance this object is immersed in to be
"infinite" to ignore its own change of temperature related to heat
emitted by our hot object.

This law of cooling should be expressed in terms of object's temperature T as a function of time t, that is, we have to find the function T(t).

Assumptions:

(a) the initial temperature of the object at time t=0 is assumed to be T₀

(b) the object has a shape of a thin flat square (so, its temperature is changing simultaneously in all its volume) of size LxL and mass m

(c) the specific heat capacity of the object's material is C

(d) the temperature of the substance surrounding our object is constant and equals T_s

(e) the convective heat transfer coefficient of the substance around our object is h.



Solution



Consider a small time interval from t to t+Δt.

The temperature of an object is T(t), while the temperature of the substance around it is constant T_s.
Then the amount of heat transferred from our object to the substance
around it per unit of time per unit of its surface area is proportional
to the difference in temperatures with the convective heat transfer coefficient as a factor:

q = −h·[T(t)−T_s]

Since total surface area of our thin flat square, ignoring its thickness, is 2L² and the time interval we consider is Δt, the total amount of heat transferred by our object through its surface during this time period is

ΔQ(t) = −2L²·h·[T(t)−T_s]·Δt

This is amount of heat taken away by the substance from our object during a time interval Δt.

The same amount of heat is lost by the object, taking its temperature from T(t) to T(t+Δt).

As we know, changes of heat and temperature are proportional and related to the object mass and specific heat capacity:

ΔQ(t) = C·m·ΔT

where ΔT = T(t+Δt)−T(t).

Equating the amount of heat lost by our object to the amount of heat
carried away by convection of the substance around it, we have come to
an equation

−2L²·h·[T(t)−T_s]·Δt =

= C·m·[T(t+Δt)−T(t)]

Dividing both parts by Δt, diminishing this time interval to zero and using a derivative by time t to express the limit, we get the following differential equation

−2L²·h·[T(t)−T_s] =

= C·m·dT(t)/dt

To solve it, let's make two simple substitutions:

(a) A = 2h·L²/(C·m)

(b) X(t) = T(t) − T_s

Then, since T_s is constant,

dT(t)/dt = dX(t)/dt

and our differential equation looks like

dX(t)/dt = −A·X(t)

To solve this, we convert it as follows:

dX(t)/X(t) = −A·dt

d[ln(X(t))] = −A·dt

Integrating:

ln(X(t))] = −A·t + B,

where B is any constant, defined by initial condition

X(0)=T(0)−T_s=T₀−T_s.

From the last equation:

X(t) = e^B·e^−A·t

Using initial condition mentioned above,

X(t) = (T₀−T_s)·e^−A·t

Since X(t)=T(t)−T_s

T(t) − T_s = (T₀−T_s)·e^−A·t

or

T(t) = T_s + (T₀−T_s)·e^−A·t

where A = 2h·L²/(C·m)

So, the difference in temperature between a hot object and infinitely
large surrounding substance is exponentially decreasing with time.