# Thin Circular Ring - Temperature and Expanding Radius

## Calculate change in pipe diameter with changed temperature

The circumference of a thin ring, a pipe or tube, can be expressed as

*c _{0} = 2 π r_{0} (1)*

*where *

*c _{0} = initial circumference (m, inches)*

*π = 3.14...*

*r_{0} = initial radius (m, inches)*

The change in circumference due to temperature change can be expressed as

*dc = c_{1 }- c_{0 }*

*_{ }* =

*2 π r*

*dt*_{0}*α (2)**where *

*dc = change in circumference (m, inches)*

*c _{1} = final circumference (m, inches)*

*dt = temperature change ( ^{o}C, ^{o}F)*

*α = linear expansion coefficient (mm/m^{o}C, μin/in^{o}F)*

The final circumference can be expressed as

*c _{1} = 2 π r_{1 } (3)*

*where *

*r_{1} = final radius (m, inches)*

Equation 1, 2 and 3 can be expressed as

*dc = *2 π r_{1} - *2 π r _{0} *

* = 2 π r_{0} dt α *

*or transformed to*

*r*_{1}* = **r*_{0}* dt α + **r*_{0}* *

* = **r*_{0}* (dt α + 1)** (4)*

Equation 4 can be modified with diameters to

*d _{1} = d_{0 }(dt *

*α + 1) (5)*

### Example - Temperature and Change in Steel Pipe Diameter

A stainless steel pipe with nominal diameter *10 inches* (outside diameter *10.750 inches*) is heated from *68 ^{o}F* to

*98*. The expansion coefficient for stainless steel S30100 is

^{o}F*9.4 μin/in*.

^{o}FThe final outside diameter can be calculated

*d _{1} = d_{0 }(dt *

*α + 1)*

* = (10.750 in) (((98 ^{o}F) - (68 ^{o}F)) (0.0000094 *

*in/in*

^{o}F) + 1)* = 10.753 inches*

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