Thermal resistance

The (absolute) heat resistance (also heat conduction resistance , thermal resistance ) is a heat parameter and a measure of the temperature difference that arises in an object when a heat flow ( heat per unit of time or heat output) occurs. The reciprocal of the thermal resistance is the thermal conductivity of the component. Dealing with resistance rather than conductance is more practical in situations where resistances appear in series , such as heat transfer to a heat sink / heat conduction in the heat sink / heat transfer to the air. ${\ displaystyle R_ {th}}$

definition

{\ displaystyle R_ {th} = {\ frac {\ Delta T} {{\ dot {Q}} _ {V}}}}

With

${\ displaystyle \ Delta T}$ - Temperature difference (e.g. between the outside and inside of a thermos bottle or between a cooling surface and the ambient air)
${\ displaystyle {{\ dot {Q}} _ {V}}}$ - Heat flow (e.g. power loss of a component mounted on a heat sink or heat lost through a window)

For a homogeneous body having on the length of constant cross-sectional area can be on the thermal conductivity calculated of the material: ${\ displaystyle l}$ ${\ displaystyle A}$ ${\ displaystyle R_ {th}}$ ${\ displaystyle \ lambda}$

{\ displaystyle R_ {th} = {\ frac {l} {\ lambda \ cdot A}}}

The unit of the thermal resistance is K / W .

Analogy to Ohm's law

Thermal quantities have analogies to those of electrical resistance, which are also shown in their names.

There are analogies to electrical current that enable the application of Ohm's law and Kirchhoff's rules to heat transfer. These are:

thermodynamics	Electrical current
Absolute thermal resistance ${\ displaystyle R_ {th} \!}$	Electrical resistance ${\ displaystyle R \!}$
Temperature difference ${\ displaystyle \ Delta T \!}$	Electric voltage ${\ displaystyle U \!}$
Heat flow ${\ displaystyle {\ dot {Q}} \!}$	Electrical current ${\ displaystyle I \!}$
Thermal conductivity ${\ displaystyle \ lambda \!}$	Electric conductivity ${\ displaystyle \ sigma \!}$
Heat capacity ${\ displaystyle C_ {th} = c_ {v} \ cdot V}$	Electrical capacitance ${\ displaystyle C}$

Application examples

Building physics

If there is a temperature difference of 20 K between the two sides of a Styrofoam sheet with a thermal resistance of 1 K / W, then the result is a heat flow through the sheet of:

{\ displaystyle {\ dot {Q}} = {\ frac {\ Delta T} {R_ {th}}} = \ mathrm {{\ frac {20 \; K} {1 \; {\ frac {K} { W}}}} = 20 \; W} \,}

Seasonal heat storage

A heat accumulator with a constant ambient temperature is discharged through its own thermal insulation . The course of the temperature difference to the environment over time is ${\ displaystyle \ Delta T}$ ${\ displaystyle t}$

{\ displaystyle \ Delta T = \ Delta T_ {0} \ cdot e ^ {- {\ frac {t} {\ tau}}}}

analogous to the voltage curve of a capacitor that is discharged through a resistor:

{\ displaystyle U = U_ {0} \ cdot e ^ {- {\ frac {t} {\ tau}}}.}

The time constant with which the heat accumulator and capacitor discharge is ${\ displaystyle \ tau}$

{\ displaystyle \ tau = R \ cdot C \ ;.}

Now as a numerical example, most of the results have been rounded: Let the heat storage medium be water with 45% ethylene glycol , 7 m wide, 7 m long, 4 m high:

{\ displaystyle V = 7 \ times 7 \ times 4 \; \ mathrm {m} ^ {3} = 196 \; \ mathrm {m} ^ {3}}

The specific heat capacity of the water-glycol mixture is

{\ displaystyle c_ {v} = 3 {,} 5 \; {\ frac {\ mathrm {MJ}} {\ mathrm {m} ^ {3} \ mathrm {K}}}}

The heat capacity is the product of volume-related specific heat capacity and volume

{\ displaystyle C _ {\ mathrm {th}} = c_ {v} \ cdot V = 686 \; \ mathrm {\ frac {MJ} {K}}}

Thermal insulation is foam glass gravel with a layer thickness . The surface of the water tank is used as the surface of the thermal insulation: ${\ displaystyle l = 0 {,} 5 \, \ mathrm {m}}$

{\ displaystyle A = (2 \ cdot 7 \ cdot 7 + 4 \ cdot 4 \ cdot 7) \; \ mathrm {m} ^ {2} = 210 \; \ mathrm {m} ^ {2}}

The specific thermal conductivity of foam glass crushed stone is

{\ displaystyle \ lambda = 0 {,} 08 \; \ mathrm {\ frac {W} {m \ K}}}

This results in thermal resistance

{\ displaystyle R _ {\ mathrm {th}} = {\ frac {l} {\ lambda \ cdot A}} = 0 {,} 03 \; \ mathrm {\ frac {K} {W}}}

Now the time constant of the self-discharge can be calculated:

{\ displaystyle \ tau = R _ {\ mathrm {th}} \ cdot C _ {\ mathrm {th}} = 20 {,} 6 \ cdot 10 ^ {6} \; \ mathrm {s} = 238 \; {\ text {days}}}

After 238 days, the difference between the temperature in the water and the surrounding area has dropped to 37% ( ) of the initial value. ${\ displaystyle = e ^ {- 1}}$

electronics

When designing the cooling of semiconductors or other circuit elements in electronic circuits , the thermal resistance of a specific heat sink is the decisive parameter for its selection. It is specified by the heat sink manufacturer, e.g. B. for free convection and should be kept as small as possible.

The thermal resistance of a component without a heat sink to the environment can be used to check whether a heat sink assembly is even necessary - it is specified by the component manufacturer as R _th J / A (from Junction / Ambient ).

In the semiconductor component itself, there is thermal resistance between the chip and the cooling surface of the housing. It is specified by the manufacturer as R _th J / C (from junction / case ).

The assembly itself and possibly a thermal pad cause further thermal resistance. If the heat sink is located inside a housing or a subrack, it must be ensured that it gives off the heat to the air, the temperature of which may be significantly higher than the temperature of the surroundings.

The temperature difference between the chip and the area around the heat sink can be calculated from the power loss and the sum of all thermal resistances : ${\ displaystyle P _ {\,} \}$ ${\ displaystyle R_ {th} \}$ ${\ displaystyle \ Delta T \}$

{\ displaystyle \ Delta T = P \ cdot R_ {th}}

If the value is too high, the heat resistance of the heat sink must be reduced, e.g. B. by a heat sink with fan or heat pipe , or to ventilate the housing.