Slabs in Series

Transfer of energy is one of the most observed phenomena in the world. Because energy can neither be created nor destroyed, it changes from one form to another. Heat is a type of energy that we all have observed in our day-to-day lives. It is also the primary type of energy that people learn to use.

The conduction of heat in all substances or slabs in series tells us how heat energy moves from one substance to another. The conduction of heat allows a rod to heat up, while the substance from which the heat is transferred loses heat.

Let us have a look at how this conduction takes place.

Conduction of heat

Conduction of heat involves the transfer of heat from one substance to another through direct collisions of subatomic particles.
As a type of energy, heat will increase the internal energy. The increase in the internal energy causes the atoms to get kinetically excited, increasing the vibrations in the body. The vibrations in the molecules at one point of the body cause the neighbouring molecules to get excited too.

In this way, the vibrations from one set of molecules are transferred to the next until it reaches the opposite end of the body. This increases the general internal energy of the complete body, and consequently, the temperature of the body also increases.

Steady state

The conduction of heat causes the temperature of the body that is subjected to heat to rise too. Conduction is similar to a chemical reaction such that, the quantity of heat coming into a frame is equal to the quantity of heat leaving the frame.

The conduction that occurs in this manner is known as steady-state conduction. It is a unique type of conduction where the temperature distribution through the area in a frame is absolutely constant. Conduction of heat is an instantaneous end result of temperature distinction among the ends of a frame. In the steady state of conduction, the temperature distinction that causes conduction no longer exists. This causes the spatial distribution of the temperature of the frame to be constant.

What this means is that if we were to pick at random a section of the body and take the temperature of the cross-section at a point that is normal to the flow of heat, then that temperature will be constant. Partial derivatives give us an idea of the changes that occur in a body. The partial derivative of heat with respect to space may yield a nonzero or a zero value, but the partial derivative of heat with respect to time that indicates a change in temperature will be zero.

A steady state of conduction is similar to the electric current in a frame. The heat that enters a frame may be assumed as heat current, and all traditional assumptions of the conduction of direct current may be applied. The temperature difference is the driving force of conduction and is similar to voltage. Here, the thermal conductivity of the frame is just like the resistance of a cord. The incoming heat is identical to the current that flows through a conductor.

Slabs in series

Let us understand the conduction of heat between two slabs that are placed in series with each other. Let us have two slabs, slab A and slab B, that are placed in series with each other. Let T1 be the temperature at one side of slab A, T2 be the temperature at the interface of the two slabs, and T3 be the temperature at the end of slab B. Let k1 be the thermal conductivity of slab A, and k2 be the thermal conductivity of slab B. Let L1 and L2 be the lengths of slab A and slab B, respectively. Let R1 and R2 be the thermal conductivities of slab A and slab B.

Since it is in steady-state conduction, the temperatures of the slabs will not change. The heat influx of both the slabs can be calculated to understand the effect that slabs in series have. Let Q be the total influx of the slabs, and Q1 and Q2 be the influx of heat in slab A and slab B, respectively.

Since the amount of heat entering the system is Q, and it is steady-state conduction, the amount of heat leaving the system is also Q. Since both the slabs are in series, the same amount of heat should also be entering slab A and slab B. Therefore,

Q = Q1 = Q2 = Q

We know,

Q = T/R where R is the total thermal conductivity of the system,

Q1 = T/R1 and Q2 = T/R2

Then,

Q = (T3 – T1)/R = (T2 – T1)/R1 = (T3 – T2)/R2

Q = (T3 – T1)/R

which means

R = R1 + R2

Therefore, for two slabs in series at steady-state conduction, the thermal conductivity of the two slabs adds up analogously to the connection of two resistors in series in an electrical circuit.

Conclusion

The conduction of heat revolves around the concept of the transfer of heat from one point to another through direct collisions of subatomic particles. The movement of electrons in atoms and the collisions of microscopic particles is the primary manner in which heat is transferred through the process of conduction.

Conduction of heat is a direct result of the temperature difference between two ends of a body. In the steady state of conduction, the temperature difference that is the driving force behind this conduction does not exist anymore. This results in the temperature field of the body or the spatial distribution of temperature of the body being constant.

For two slabs in series at steady-state conduction, the thermal conductivity of the two slabs adds up analogously to the connection of two resistors in series in an electrical circuit.