A temperature controller, designed to work in a steam...

A temperature controller, designed to work in a steam environment, involves a bimetallic strip constructed of brass and steel, connected at their ends by rivets. Each of the metals is 2.0 $\mathrm{mm}$ thick. At $20^{\circ} \mathrm{C}$ the strip is 10.0 $\mathrm{cm}$ long and straight. Find the radius of curvature $r$ of the assembly at $100^{\circ} \mathrm{C}$ . See Fig. $22 .$

A temperature controller, designed to work in a steam environment, involves a bimetallic strip constructed of brass and steel, connected at their ends by rivets. Each of the metals is 2.0 $\mathrm{mm}$ thick. At $20^{\circ} \mathrm{C}$ the strip is 10.0 $\mathrm{cm}$ long and straight.
Find the radius of curvature $r$ of the assembly at $100^{\circ} \mathrm{C}$ . See Fig. $22 .$

Key Concepts

Thermal Expansion

Thermal expansion is the phenomenon by which materials change their dimensions in response to a change in temperature. Each material has a characteristic coefficient of thermal expansion, which quantifies how much it expands or contracts per degree change in temperature. This concept is fundamental in understanding how temperature variations can induce mechanical stresses in materials and assemblies.

Differential Thermal Expansion

Differential thermal expansion occurs when two materials with different coefficients of thermal expansion are joined together, causing them to expand or contract by different amounts when subjected to a temperature change. This mismatch in expansion leads to bending or warping in composite structures, as the material that expands more imposes a strain on the material that expands less.

Bimetallic Strip

A bimetallic strip is a composite element made by bonding two strips of different metals together. Due to the differing thermal expansion properties of the two metals, changes in temperature cause the strip to bend. This bending is a direct consequence of the imposed differential expansion, and bimetallic strips are widely used for temperature sensing and control applications.

Beam Bending and Curvature

Beam bending theory deals with how beams deform under applied loads or due to internal stresses such as thermal expansion. One of the key parameters in this context is the radius of curvature, which quantifies the extent of bending. The concepts of the neutral axis and strain distribution along the beam cross-section are essential in relating the bending moment or differential strain to the curvature of the beam.

Transcript

00:01 A temperature controller designed to work in a steam environment involves a bimetallic strip, constructive brass and steel connected at their ends by rivets.

00:10 Each of the metal is two millimetres thick.

00:14 At 20 degrees celsius, the strip is 10 centimetres long and straight.

00:19 Find the radius of curvature are of the assembly at 100 degrees.

00:22 Okay, so this is a reasonably challenging problem, but let's go on with it.

00:28 So looking at the thermal expansion formula for a metal, delta l, or for any substance for that matter, equals, so we're going to look at delta ls for steel, changing length of steel, is equal to alpha s, and we know that the changing temperature, so delta t, so alpha delta t current length.

00:53 So we know that our change temperature is 80 here, 80, and our, so multiplied by 80, and our current length, we're going to work in centimeters for the minute because it's just a little bit easier times by 10.

01:12 We know that alpha s, the coefficient for still, is equal to 12 times 10 to minus 6.

01:24 And similarly for bross, delta lb is equal to alpha b.

01:33 So coefficient of expansion for brass times by 80 times by 10.

01:38 And we are going to say that our coefficient for brass is equal to 19 times 10 to the minus 6.

01:51 And these units are degrees celsius to the minus 1.

01:59 And substituting in those numbers, sorry, that should be 80 times 10.

02:06 Sorry.

02:09 And so substituting in those numbers, we get that.

02:14 So we do, in order to get the length, we do ls plus, so 10 plus delta ls.

02:23 It turns out the delta ls is equal to, so how on, and delta l s is equal to 10, sorry, is equal to 0 .0 .996 centimeters.

02:43 And delta lb, so the changing length of the brass, is equal to 0 .0152 centimeters.

02:57 So starting a new page here, that gives us the new length of steel ls is equal to 10 .0.

03:13 So this is 100 degrees.

03:15 The length of the steel beam is 10 .0 .096 centimeters.

03:21 And the length of the brass beam is equal to a 10 .0 .0...

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