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# Joule heating

Process by which the passage of an electric current through a conductor produces heat

A coiled

heating element

from an electric toaster, showing red to yellow

incandescence

Joule heating, also known as resistive, resistance, or Ohmic heating, is the process by which the passage of an

electric current

through a

conductor

produces

heat

.

Joule’s first law, also known as the Joule–Lenz law,

[1]

states that the

power

of heating generated by an

electrical conductor

is proportional to the product of its

resistance

and the square of the current:

P∝I2R{displaystyle Ppropto I^{2}R}

Joule heating affects the whole electric conductor, unlike the

Peltier effect

which transfers heat from one electrical junction to another.

## History

James Prescott Joule

first published in December 1840, an abstract in the

Proceedings of the Royal Society

, suggesting that heat could be generated by an electrical current. Joule immersed a length of wire in a fixed

mass

of

water

and measured the

temperature

rise due to a known current flowing through the wire for a 30

minute

period. By varying the current and the length of the wire he deduced that the heat produced was

proportional

to the

square

of the current multiplied by the

electrical resistance

of the immersed wire.

[2]

In 1841 and 1842, subsequent experiments showed that the amount of heat generated was proportional to the

chemical energy

used in the

voltaic pile

that generated the template. This led Joule to reject the

caloric theory

(at that time the dominant theory) in favor of the

mechanical theory of heat

(according to which heat is another form of

energy

).

[2]

Resistive heating was independently studied by

Heinrich Lenz

in 1842.

[1]

The

SI unit

of

energy

was subsequently named the

joule

and given the symbol J. The commonly known unit of power, the

watt

, is equivalent to one joule per second.

## Microscopic description

Joule heating is caused by interactions between

charge carriers

(usually

electrons

) and the body of the conductor (usually

atomic

ions

).

A

voltage

difference between two points of a conductor creates an

electric field

that accelerates charge carriers in the direction of the electric field, giving them

kinetic energy

. When the charged particles collide with ions in the conductor, the particles are

scattered

; their direction of motion becomes random rather than aligned with the electric field, which constitutes

thermal motion

. Thus, energy from the electrical field is converted into

thermal energy

.

[3]

## Power loss and noise

Joule heating is referred to as ohmic heating or resistive heating because of its relationship to

Ohm’s Law

. It forms the basis for the large number of practical applications involving

electric heating

. However, in applications where heating is an unwanted

by-product

of current use (e.g.,

in

electrical transformers

) the diversion of energy is often referred to as resistive loss. The use of

high voltages

in

electric power transmission

systems is specifically designed to reduce such losses in cabling by operating with commensurately lower currents. The

ring circuits

, or ring mains, used in UK homes are another example, where power is delivered to outlets at lower currents (per wire, by using two paths in parallel), thus reducing Joule heating in the wires. Joule heating does not occur in

superconducting

materials, as these materials have zero electrical resistance in the superconducting state.

Resistors create electrical noise, called

Johnson–Nyquist noise

. There is an intimate relationship between Johnson–Nyquist noise and Joule heating, explained by the

fluctuation-dissipation theorem

.

## Formulas

### Direct current

The most fundamental formula for Joule heating is the generalized power equation:

P=I(VA−VB){displaystyle P=I(V_{A}-V_{B})}

where

• P{displaystyle P} is the

power

(energy per unit time) converted from electrical energy to thermal energy,

• I{displaystyle I} is the current travelling through the resistor or other element,
• VA−VB{displaystyle V_{A}-V_{B}} is the

voltage drop

across the element.

The explanation of this formula (P=IV{displaystyle P=IV}) is:

[4]

(Energy dissipated per unit time) = (Charge passing through resistor per unit time) × (Energy dissipated per charge passing through resistor)

Assuming the element behaves as a perfect resistor and that the power is completely converted into heat, the formula can be re-written by substituting

Ohm’s law

, V=I⋅R{displaystyle V=Icdot R}, into the generalized power equation:

P=IV=I2R=V2/R{displaystyle P=IV=I^{2}R=V^{2}/R}

where R is the

resistance

.

### Alternating current

When current varies, as it does in AC circuits,

P(t)=U(t)I(t){displaystyle P(t)=U(t)I(t)}

where t is time and P is the instantaneous power being converted from electrical energy to heat. Far more often, the average power is of more interest than the instantaneous power:

Pavg=UrmsIrms=Irms2R=Urms2/R{displaystyle P_{rm {avg}}=U_{text{rms}}I_{text{rms}}=I_{text{rms}}^{2}R=U_{text{rms}}^{2}/R}

where “avg” denotes

average (mean)

over one or more cycles, and “rms” denotes

root mean square

.

These formulas are valid for an ideal resistor, with zero

reactance

. If the reactance is nonzero, the formulas are modified:

Pavg=UrmsIrmscos⁡ϕ=Irms2Re⁡(Z)=Urms2Re⁡(Y∗){displaystyle P_{rm {avg}}=U_{text{rms}}I_{text{rms}}cos phi =I_{text{rms}}^{2}operatorname {Re} (Z)=U_{text{rms}}^{2}operatorname {Re} (Y^{*})}

where ϕ{displaystyle phi } is phase difference between current and voltage, Re{displaystyle operatorname {Re} } means

real part

, Z is the

complex impedance

, and Y* is the

complex conjugate

of the

(equal to 1/Z*).

For more details in the reactive case, see

AC power

∆0}

### Differential form

Joule heating can also be calculated at a particular location in space. The differential form of the Joule heating equation gives the power per unit volume.

dP/dV=J⋅E{displaystyle mathrm {d} P/mathrm {d} V=mathbf {J} cdot mathbf {E} }

Here, J{displaystyle mathbf {J} } is the current density, and E{displaystyle mathbf {E} } is the electric field. For a material with a conductivity σ{displaystyle sigma }, J=σE{displaystyle mathbf {J} =sigma mathbf {E} } and therefore

dP/dV=J⋅E=J⋅=J2/σ{displaystyle mathrm {d} P/mathrm {d} V=mathbf {J} cdot mathbf {E} =mathbf {J} cdot mathbf {J} rho =J^{2}/sigma }

where ρ=1/σ{displaystyle rho =1/sigma } is the

resistivity

. This directly resembles the “I2R{displaystyle I^{2}R}” term of the macroscopic form.

In the harmonic case, where all field quantities vary with the angular frequency ω{displaystyle omega } as e−t{displaystyle e^{-mathrm {i} omega t}}, complex valued

phasors

J^{displaystyle {hat {mathbf {J} }}} and E^{displaystyle {hat {mathbf {E} }}} are usually introduced for the current density and the electric field intensity, respectively. The Joule heating then reads

dP/dV=12J^E^=12J^J^ρ=12J2/σ{displaystyle mathrm {d} P/mathrm {d} V={frac {1}{2}}{hat {mathbf {J} }}cdot {hat {mathbf {E} }}^{*}={frac {1}{2}}{hat {mathbf {J} }}cdot {hat {mathbf {J} }}^{*}rho ={frac {1}{2}}J^{2}/sigma },

where {displaystyle bullet ^{*}} denotes the

complex conjugate

.

## High-voltage alternating current transmission of electricity

transfer electrical energy from electricity producers to consumers. Those power lines have a nonzero resistance and therefore are subject to Joule heating, which causes transmission losses.

The split of power between transmission losses (Joule heating in transmission lines) and load (useful energy delivered to the consumer) can be approximated by a

voltage divider

. In order to minimize transmission losses, the resistance of the lines has to be as small as possible compared to the load (resistance of consumer appliances). Line resistance is minimized by the use of

copper conductors

, but the resistance and

power supply

specifications of consumer appliances are fixed.

Usually, a

transformer

is placed between the lines and consumption. When a high-voltage, low-intensity current in the primary circuit (before the transformer) is converted into a low-voltage, high-intensity current in the secondary circuit (after the transformer), the equivalent resistance of the secondary circuit becomes higher

[5]

and transmission losses are reduced in proportion.

During the

war of currents

,

AC

installations could use transformers to reduce line losses by Joule heating, at the cost of higher voltage in the transmission lines, compared to

DC

installations.

## Applications

Joule-heating or resistive-heating is used in multiple devices and industrial process. The part which converts electricity into heat by Joule heating is called a

heating element

.

There are many practical uses of Joule heating:

• An

incandescent light bulb

glows when the filament is heated by Joule heating, due to

(also called

).

• Electric fuses

are used as a safety, breaking the circuit by melting if enough current flows to melt them.

• Electronic cigarettes

vaporize propylene glycol and vegetable glycerine by Joule heating.

• Multiple heating devices use Joule heating, such as

electric stoves

,

electric heaters

,

soldering irons

,

cartridge heaters

.

• Some

food processing

equipment may make use of Joule heating: running current through food material (which behave as an electrical resistor) causes heat release inside the food.

[6]

The alternating electrical current coupled with the resistance of the food causes the generation of heat.

[7]

A higher resistance increases the heat generated. Ohmic heating allows for fast and uniform heating of food products, which keeps the high quality in foods. Products with particulates heat up faster in Ohmic heating (as compared to conventional heat processing) due to higher resistance.

[8]

### Food processing

Joule heating (

Ohmic heating

) is a

flash pasteurization

(also called “high-temperature short-time” (HTST)) aseptic process that runs an alternating current of 50–60 Hz through food.

[9]

Heat is generated through the electrical resistance of the food.

[9]

As the product heats up, electrical conductivity increases linearly.

[7]

A higher electrical current frequency is best as it reduces oxidation and metallic contamination.

[9]

This heating method is best for foods that contain particulates suspended in a weak salt-containing medium due to their high resistance properties.

[8]

Ohmic heating allows for a maintained quality of foods due to the uniform heating that decreases deterioration and over-processing of food.

[9]

## Heating efficiency

As a heating technology, Joule heating has a

coefficient of performance

of 1.0, meaning that every joule of electrical energy supplied produces one joule of heat. In contrast, a

heat pump

can have a coefficient of more than 1.0 since it moves additional thermal energy from the environment to the heated item.

The definition of the efficiency of a heating process requires defining the boundaries of the system to be considered. When heating a building, the overall efficiency is different when considering heating effect per unit of electric energy delivered on the customer’s side of the meter, compared to the overall efficiency when also considering the losses in the power plant and transmission of power.

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## Hydraulic equivalent

In the

energy balance of groundwater flow

a hydraulic equivalent of Joule’s law is used:

[10]

dEdx=vx2K{displaystyle {dE over dx}={v_{x}^{2} over K}}

where:

dE/dx{displaystyle dE/dx} = loss of hydraulic energy (E{displaystyle E}) due to friction of flow in x{displaystyle x}-direction per unit of time (m/day) – comparable to P{displaystyle P}
vx{displaystyle v_{x}} = flow velocity in x{displaystyle x}-direction (m/day) – comparable to I{displaystyle I}
K{displaystyle K} =

hydraulic conductivity

of the soil (m/day) – the hydraulic conductivity is inversely proportional to the hydraulic resistance which compares to R{displaystyle R}

• Resistance wire

• Heating element

• Nichrome

• Tungsten

• Molybdenum disilicide

• Overheating (electricity)

• Thermal management (electronics)

• Induction heating

## References

1. ^

a

b

Джоуля — Ленца закон

Archived

2014-12-30 at the

Wayback Machine

. Большая советская энциклопедия, 3-е изд., гл. ред. А. М. Прохоров. Москва: Советская энциклопедия, 1972. Т. 8 (A. M. Prokhorov; et al., eds. (1972). “Joule–Lenz law”.

Great Soviet Encyclopedia

(in Russian). 8. Moscow: Soviet Encyclopedia.)

2. ^

a

b

“This Month Physics History: December 1840: Joule’s abstract on converting mechanical power into heat”

. aps.org. American Physical society. Retrieved 16 September 2016.

3. ^

“Drift Velocity, Drift Current and Electron Mobility”

. Electrical4U. Retrieved 26 July 2017.

4. ^

Electric power systems: a conceptual introduction by Alexandra von Meier, p67,

5. ^

“Transformer circuits”

. Retrieved 26 July 2017.

6. ^

Ramaswamy, Raghupathy.

“Ohmic Heating of Foods”

. Ohio State University. Archived from

the original

on 2013-04-08. Retrieved 2013-04-22.

7. ^

a

b

Fellows, P.J (2009). Food Processing Technology. MA: Elsevier. pp. 813–844.

ISBN

978-0-08-101907-8

.

8. ^

a

b

Varghese, K. Shiby; Pandey, M. C.; Radhakrishna, K.; Bawa, A. S. (October 2014).

“Technology, applications and modelling of ohmic heating: a review”

. Journal of Food Science and Technology. 51 (10): 2304–2317.

doi

:

10.1007/s13197-012-0710-3

.

ISSN

0022-1155

.

PMC

4190208

.

PMID

25328171

.

9. ^

a

b

c

d

1953-, Fellows, P. (Peter) (2017) [2016]. Food processing technology : principles and practice (4th ed.). Kent: Woodhead Publishing/Elsevier Science.

ISBN

9780081019078

.

OCLC

960758611

.CS1 maint: numeric names: authors list (

)

10. ^

R.J.Oosterbaan, J.Boonstra and K.V.G.K.Rao (1996).

The energy balance of groundwater flow

(PDF). In: V.P.Singh and B.Kumar (eds.), Subsurface-Water Hydrology, Vol.2 of the Proceedings of the International Conference on Hydrology and Water Resources, New Delhi, India. Kluwer Academic Publishers, Dordrecht, The Netherlands. pp. 153–160.

ISBN

978-0-7923-3651-8

.