## Thursday, April 24, 2008

### Concept review Ch. 28 Heat transfer

Revision points

Heat can be transferred from one place to another by three different methods.

Conduction
Convection
Radiation

Conduction takes place in metals.

Convection takes place liquids and gases.

No medium is required for radiation

Thermal conduction

In a metal rod, if one end is heated, the molecules at that end gain heat and their average kinetic energy increases. They collide with neighbouring molecules which have less kinetic energy and the energy sharing takes place. The molecules which now gained some additional energy in turn collide with their neighbour at lower energy and share energy with them. This way, heat is passed along the rod from one end to the other end from molecule to molecule.

The average position of any molecule does not change. Hence there is no mass movement in thermal conduction.

Thermal conductivity is a measure of the ability of a material to conduct heat.

Heat current: In steady state, if ΔQ amount of heat crosses through any cross section material in time Δt, ΔQ/Δt is called the heat current.

In steady state, heat current is proportional to the area of cross section A, proportional to the temperature differences between the ends of body (for example metal rod) and inversely proportional to the length if area of cross section is uniform throughout the body.

Heat current between two ends = ΔQ/Δt = KA(T1 – T2)/x

Where
K = constant for the material of the body and is called the thermal conductivity
Units: J/s-m-K or W/m-K we can also write °C in place of K.
A = area of cross section
T1, T2) = higher and lower temperatures of ends.
x = distance between ends

If the area of cross section is not uniform, the formula is applied to small thicknesses only.
ΔQ/Δt = - KAdT/dx

The quantity dT/dx is called the temperature gradient. The minus sign indicates that in the direction of heat flow there has to be drop in temperature.

In solids metals are better conductors than nonmetals. Metals have free electrons that move in the body of the metal freely and help in carrying thermal energy from one place to another.

Thermal conductivities

Aluminium 209 K(W/m-K)
Glass 0.669 K(W/m-K)

When a rug is placed in bright sun on a tiled floor, one can easily stand on rug. But one cannot stand on the bare floor. Reason: Both rug and floor are at the same temperature. No heat transfer takes place between surfaces at the same temperature. But the rug has very low conductivity and therefore heat current going into the foot is small when one is standing on rug.

Thermal resistance: The quantity x/KA in the equation for ΔQ/Δt is termed the thermal resistance R. So we can write

ΔQ/Δt = KA(T1 – T2)/x = (T1 – T2)/R

We can write ΔQ/Δt as i heat current and then write i = (T1 – T2)/R.

This formula is equivalent to the Ohm’s law formula of electric current. Many results derive based on the Ohm’s law for electric will be applicable to the heat current formula also.

When thermal resistances of same cross section are joined in series, just as in an electrical circuit current in same in the circuit and in both resistances, in the thermal resistances also the heat current will be same.

Hence the equivalent thermal resistance when thermal restances R1 and R2 are connected in series is R1+R2.

Heat current in the system will be i = T1-T2/R1+R2

When thermal resistances are connected in parallel, that is both the left ends of the two metal rods are a temperature T1 and connected to the same heat source and both right ends of the two metal rods are at the same temperature T2

i1 = T1-T2/R1

i2 = T1-T2/R2

i = i1 + i2 = T1-T2[1/R1 + 1/R2]

Hence equivalent resistance R = 1/R1 + 1/R2

If many thermal resistances are connected in parallel, equivalent resistance

R = 1/R1 + 1/R2 + 1/R3 ...

Measurement of Thermal Conductivity of a Solid

Searle’s apparatus is used for measuring thermal conductivity of a solid. One end of the cylindrical rod of a metal whose thermal conductivity is to be measured is placed in steam chamber. At the other end water circulates through a copper tube whose inlet and outlets have thermometers to measure inlet (θ4) and outlet (θ3) temperatures. In between, on the rod there are two drilled holes filled with mercury and thermometers are placed in these holes to measure temperatures (θ1 and θ2, θ1 is greater than θ2). After the steam is passed into the steam chamber, sufficient time is allowed for the system to stabilize, that is all temperatures are constant. Then for a measured t water is collected in a flask and weighed.

Heat flow between holes having thermometers = KA(θ1 – θ2)t/x

Where
K = thermal conductivity of the material
A = area of cross section of cylindrical rod
x = distance between the holes (thermometers)

The heat flow into water collected = ms(θ3 – θ4)
m = mass of water collected
s = specific heat capacity of water

Both the heat flows are same and hence KA(θ1 – θ2)t/x = ms(θ3 – θ4)
K = [x ms(θ3 – θ4)]/ [A(θ1 – θ2)t]

Convection

In convection heat transferred from one place to the other side by the actual motion of heated material. When water is kept in a vessel and heated on a stove, the water at the bottom gets heat due to conduction through the vessels bottom. Its density decreases and consequently it rises. Hot molecules keep going up and cold molecules keep coming down. Mass transfer accomplishes heat transfer from part to the other part of the liquid.

If the heated material is forced to move using a pump, it is called forced convection.

The movement of liquid in heat transfer and anomalous expansion of water (it expands as temperature falls from 4°C to 0°C.) saves the lives of acquatic animal during severe winters. During winters, as the surface of water gets cool, the water at the surface becomes dense and goes inside and hot water from inside comes up. But after 4°C, water at the surface does not become dense, it expands and hence water at the surface stays at the surface and freezes into ice. Heat transfer through ice is a very slow process. Therefore water is there under freezed ice at the surface acquatic animals survive in the water.

Radiation

Radiation word has two meanings. One, the energy transmitted by radiation process is called radiation. The other is the radiation process. In radiation process, heat transfer takes place without the involvement of any medium. Energy emitted by a body travels in the space and when it falls on material body, a part of it is absorbed and the thermal energy of the receiving body increases.

Prevost Theory of Exchange

According to this theory all bodies radiate at all temperatures. The amount of thermal radiation radiated per unit time depends on the nature of the emitting surface, its area and its temperature.

Every body absorbs part of the thermal radiation emitted by the surrounding bodies when this radiation falls on it.

If a body radiates more than what it absorbs, its temperature falls.

Kirchoff’s law

The ratio of emissive power to absorptive power is the same for all bodies at a given temperature and is equal to the emissive power of a blackbody at that temperature.

Thus
E(body)/a(body) = E(blackbody)

Stefan-Boltzmann Law

The energy of thermal radiation emitted by per unit time by a black body of surface area A is given by

U = σAT4

Where
σ = Stefan Boltzmann constant = 5.67*10-8 W/m²-K4

Newton’s law of cooling

dT/dt = -bA(T-T0)

b = a constant depends on the nature of the surface involved
a = surface area exposed of the body
T- T0) = temperature difference between the body and surrounding

Detection and Measurement of Radiation

Bolometer and thermopile are used to detect and measure thermal radiation.