What are common results of heat transfer?

Case #1, no phase transition or work done. How much does the temperature vary?

Heat is energy in transit! Positive, when obtained.

1. Growth of temperature.

2. A phase transition (melting ice).

3. Mechanical work.

In cases #2 and #3 there may be NO temperature variation.

Heat capacity of the object C, measured in J/K; tells you how much Joules of heat you need to transfer to increase the temperature of the object by 1 K (or 1 ºC). T

QC

TCQ

T

QC

The term “heat capacity” is used for historic reasons and is confusing!

It sounds as if the object “contains heat”, whereas by definition heat is energy in transit. An object contains some internal energy (not heat!) and its temperature is a measure of this internal energy.

An analogy from mechanics: you do some work on an object and the object gains the same amount of energy (kinetic, potential) as a result.

Mechanical work also relates to a process, not state.

WhmgK

Heat capacity of the object C, measured in J/K; tells you how much Joules of heat you need to transfer to increase the temperature of the object by 1 K (or 1 ºC).

Heat capacity is an extensive (integral) parameter:When you bring two objects together, heat capacity of the system of the two objects becomes the sum of the two individual heat capacities.

Tm

Q

m

Cc

TcmQ

It is convenient to introduce specific heat, c, which is heat capacity of a material per unit mass.

Specific heat is measured in J/(Kkg).

Heat capacity of a water balloon is large due to both high specific heat and large mass of the water inside the balloon.

cmC

Tm

Q

m

Cc

TcmQ

Specific heat, c, is heat capacity of a material per unit mass.

Specific heat is measured in J/(Kkg).

021 QQ

The equilibrium temperature.

Situation: two objects with different temperatures, T1 and T2, are

brought in a thermal contact and reach thermal equilibrium at a

temperature T after a while.Heat Q1 is transferred to Object 1; heat Q2 is transferred to Object 2.

By energy conservation:

21 QQ By definition of heat capacity and specific heat:

)( 111111111 TTcmTcmTCQ )( 222222222 TTcmTcmTCQ

0)()( 222111 TTcmTTcm

2211

222111

cmcm

TcmTcmT

21

2211

CC

TCTC

Heat transfer

1.Conduction.

2. Convection

3. Radiation

the area of the slab, the

temperature difference, T, between the back and the front and inversely proportional

Setting:

A rectangular slab of thickness x

and with an area A.

The front side of the slab is at a

temperature T; the back side has a somewhat

different temperature, T+T.

We are trying to calculate the heat-flow rate, the amount of heat flowing through the slab per unit time,

H = Q/t.

to the thickness of the slab, x.

H should also somehow depend of properties of the material the slab is made of…

We expect H to be proportional to

Bringing all the parts together:

H = Q/t – heat-flow rate is measured in Joules/second, J/s, or Watts, W.

Thermal conductivity, k, is measured in W/(mK).

x

TkAH

The coefficient k reflects specific properties of the material of the slab and is called thermal conductivity

Thermal conductivities of different materials.

Best heat conductor – Copper; use it when you build heat sink, as a material for pipes in your cooling system, a radiator.

Worst heat conductors are the best insulating materials – air, fiberglass (layers in the walls of houses in cold regions), styrofoam (cups for your hot coffee).

Heat-flow rate equation

Continuing the analogy: electric resistance, R, is analogous to

)/(kAx

T

x

TkAH

Is similar to the Ohm’s law:

R

VI

The current, I = q/t, amount of charge per unit time,

is analogous to the heat-flow rate, H = Q/t. The voltage, V, the factor driving the electric current, is analogous to temperature difference, T.

kA

x

Thermal resistance is introduced askA

xR

Heat-flow rate equationR

TH

Thermal resistance is introduced askA

xR

A composite slab is analogous to two

resistors connected in a series.

Heat-flow rate

21

13

total

total

RR

TT

R

TH

Ak

xR

1

11

Ak

xR

2

22

Reducing heat-flow rate for better thermal insulation.

Better thermal insulation… red or blue?

Licking an ice cream, which is frozen, seems to be OK…

What about the door handles, when it is freezing outside?

Which one would you prefer?

Cold metals are especially bad because of their high… thermal conductivity

In order to keep your tongue above 0 °C you basically have to heat the whole piece of metal…

Otherwise…

Thermal conductivities of different materials.

Sample problem

A home heating system supplies heat at the maximum rate of 40 kW. If the house loses 1.1 kW for each °C between inside and outside, what is the minimal outdoor temperature for which the heating system can maintain 20 °C inside?

kW401

kW1.1max

HT

CHout

Solution: the lowest temperature outside for which the heating system can maintain 20 °C inside corresponds to the case, when the

maximal power of the heater, Hmax=40 kW all flows outside because of temperature difference.

CCT 36kW1.1

kW40CTTT inout 16

Convection - Heat transfer in a gas or liquid by the circulation of currents from one region to another.Can be forced or spontaneous (natural).

Hot and cold liquid is brought in a thermal contact; it reduces the distance across which the conduction occurs and increases the contact area.

x

TkAH