linear thermal expansion coefficient

(noun)

The fractional change in length per degree of temperature change.

Related Terms

Examples of linear thermal expansion coefficient in the following topics:

  • Area Expansion

    • We learned about the linear expansion (in one dimension) in the previous Atom.
    • The area thermal expansion coefficient relates the change in a material's area dimensions to a change in temperature.
    • This equation works well as long as the linear expansion coefficient does not change much over the change in temperature $\Delta T$.
    • For isotropic materials, and for small expansions, the linear thermal expansion coefficient is one half of the area coefficient.
    • Express the area thermal expansion coefficient in the form of an equation

  • Volume Expansion

    • The volumetric thermal expansion coefficient is the most basic thermal expansion coefficient. illustrates that, in general, substances expand or contract when their temperature changes, with expansion or contraction occurring in all directions.
    • For isotropic materials, the area and linear coefficients may be calculated from the volumetric coefficient (discussed below).
    • For a solid, we can ignore the effects of pressure on the material, thus the volumetric thermal expansion coefficient can be written:
    • For isotropic material, and for small expansions, the linear thermal expansion coefficient is one third the volumetric coefficient.
    • (and from the definitions of the thermal coefficients), we arrive at:

  • Linear Expansion

    • Thermal expansion is the tendency of matter to change in volume in response to a change in temperature.
    • The degree of expansion divided by the change in temperature is called the material's coefficient of thermal expansion; it generally varies with temperature.
    • To a first approximation, the change in length measurements of an object (linear dimension as opposed to, for example, volumetric dimension) due to thermal expansion is related to temperature change by a linear expansion coefficient.
    • From the definition of the expansion coefficient, the change in the linear dimension $\Delta L$ over a temperature range $\Delta T$ can be estimated to be:
    • This equation works well as long as the linear-expansion coefficient does not change much over the change in temperature.

  • Thermal Stresses

    • Solids also undergo thermal expansion.
    • where ΔL is the change in length L, ΔT is the change in temperature, and α is the coefficient of linear expansion, which varies slightly with temperature.
    • Thermal stress is created by thermal expansion or contraction.
    • One challenge is to find a coating that has an expansion coefficient similar to that of metal.
    • If the expansion coefficients are too different, the thermal stresses during the manufacturing process lead to cracks at the coating-metal interface.

  • Dependence of Resistance on Temperature

    • Resistivity and resistance depend on temperature with the dependence being linear for small temperature changes and nonlinear for large.
    • where ρ0 is the original resistivity and α is the temperature coefficient of resistivity.
    • They become better conductors at higher temperature because increased thermal agitation increases the number of free charges available to carry current.
    • (Examination of the coefficients of linear expansion shows them to be about two orders of magnitude less than typical temperature coefficients of resistivity, and so the effect of temperature on L and A is about two orders of magnitude less than on ρ. ) Thus,
    • The device is small so it quickly comes into thermal equilibrium with the part of a person it touches.

  • Temperature

    • These basic measuring tools utilized the expansion and contraction of air and water when heated and cooled.
    • Fahrenheit was working with tubes filled with mercury, which has a very high coefficient of thermal expansion.
    • The bimetallic strips are made from two dissimilar metals bonded together, with each metal having a different coefficient of thermal expansion.
    • The fundamental requirements of the practice involve accuracy, a standard, linearity, and reproducibility.

  • Determinants of 2-by-2 Square Matrices

    • The determinant of a square matrix is computed by recursively performing the Laplace expansion to find the determinant of smaller matrices.
    • A matrix is often used to represent the coefficients in a system of linear equations, and the determinant can be used to solve those equations.
    • Determinants are also used to define the characteristic polynomial of a matrix, which is essential for eigenvalue problems in linear algebra.
    • In linear algebra, the determinant is a value associated with a square matrix.

  • Superposition and orthogonal projection

    • where $X$ is the matrix whose columns are the $\mathbf{x}_i$$\mathbf{c}$ vectors and $\mathbf{c}$ is the vector of unknown expansion coefficients.
    • Suppose we are trying to find the coefficients of
    • In this case we can find the coefficients easily by projecting onto the orthogonal directions:
    • In general, the sum will require an infinite number of coefficients $c_i$ , since a function has an infinite amount of information.
    • Using Equations 4.3.8 and 4.3.9 we can compute the Fourier coefficients by simply projecting $f(x)$ onto each orthogonal basis vector:

  • Partial Fractions

    • The main motivation to decompose a rational function into a sum of simpler fractions is to make it easier to perform linear operations on the sum.
    • To complete the process, we must determine the values of these $c_i$ coefficients. 
    • To find a coefficient, multiply the denominator associated with it by the rational function $R(x)$:
    • Substituting these coefficients into the decomposed function, we have:
    • We have solved for each constant and have our partial fraction expansion: