Conduction - A118
Conduction | |
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Foundational knowledge article | |
Document Type | Article |
Document Identifier | 118 |
Themes | |
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Material |
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Prerequisites |
Introduction[edit | edit source]
Conduction is a molecular heat transfer mode associated with the microscopic collisions between high energy particles and adjacent lower energy particles causing a redistribution of kinetic energy. Therefore, heat transfer by conduction occurs from regions of higher temperature to regions of lower temperatures.
Scope[edit | edit source]
This page explains thermal conduction in the context of composites processing and how it differs from other heat transfer mechanisms such as convection. Specifics about the material properties involved in thermal conduction are given in separate pages.
Significance[edit | edit source]
Conduction is the primary heat transfer mechanism in solid materials. It plays a critical role in composite manufacturing processes, responsible for heat flow in hot press forming platens, tooling and part through-thickness temperature gradients. Both steady-state and transient (non steady-state) conduction are introduced and discussed.
Prerequisites[edit | edit source]
Recommended documents to review before, or in parallel with this document:
Conduction overview[edit | edit source]
Conduction is a molecular heat transfer mode associated with the microscopic collisions between high energy particles and adjacent lower energy particles causing a redistribution of kinetic energy. Therefore, heat transfer by conduction occurs from regions of higher temperature to regions of lower temperatures. It is one of the are three main mechanisms of heat transfer.
Using the classic example of a metal rod being heated over a flame; the heat being transferred from the flame to the rod is considered to be by convection, while the heat transferred along the rod itself is by conduction ^{[1]}. Applying this analogy to the composites processing scenario for a laminate curing in an oven or autoclave environment; the hot air transfers heat by convection, while the heat transfer between the tool and laminate part, and within the laminate itself is by conduction.
There are two main properties to heat conduction:
Thermal conduction occurs under two conditions: steady-state, and non-steady or transient conditions.
Steady-state conduction[edit | edit source]
Steady-state conduction is described empirically by Fourier's Law.
\(\overrightarrow{q}=-k\overrightarrow{\bigtriangledown}T\)
Where,
\(\overrightarrow{q}=\)heat flux
\(k=\) thermal conductivity
\(\overrightarrow{\bigtriangledown}T=\)temperature gradient
Fourier’s Law provides an empirical description of conduction stating that the heat flux, q, travelling from a region of higher temperature to a region of lower temperature is equal to the product of the temperature gradient, ΔT, between the two regions and the thermal conductivity of the material, k. This means that the energy transport by conduction between two regions increases with the temperature difference between the two regions. A simple one-dimensional heat transfer case by solid-state conduction is shown with a wall exposed on one side to a hot fluid and to a cold fluid on the other side. Under steady state conditions a linear temperature gradient develops inside the wall as heat is transferred from the hot fluid through the solid wall to the cold fluid via conduction.
For a simple steady-state one-dimensional case, the differential form of Fourier's Law simplifies to:
Which, if the temperature distribution is linear, becomes:
Where \(q_x\) is the heat flux in [W/m^{2}] in the x direction, \(\bigtriangleup T\) is the temperature gradient in [K/m] in the \(x\) direction, and \(k\) is the thermal conductivity of the material in [W/m·K].
In order to understand the 1D steady-state temperature distribution through a part, where one side side is hotter than the other, the differential form of q_{x} must be substituted into the fundamental energy equation. Take the following example, where an infinitely long wall is heated from one side.
\(-\frac{\partial q}{\partial x}-\frac{\partial q}{\partial y}-\frac{\partial q}{\partial z}+\dot Q_{gen}-\dot Q_{cons}=\rho c_p\frac{\partial T}{\partial t}\)
For this case, the one-dimensional form of the energy equation, with no internal heat generation (\(\dot Q_{gen}\)) or accumulation (\(\dot Q_{cons}\)), i.e., steady state, becomes:
Which gives:
Integrating the above equation and solving for the boundary conditions (i.e. T(0) = T_{0} and T(L) = T_{1}) yields:
Under steady-state conditions, the temperature profile increases linearly from T_{0} to T_{1}.
Transient conduction[edit | edit source]
Transient (non-steady state) conduction involves heat capacity (\(c_p\)), density (\(\rho\)), thermal diffusivity (\(\alpha\)), and a rate term.
The thermal diffusivity is defined as\[\alpha=\frac{k}{\rho c_p}\]
Where,
\(k=\)thermal conductivity [W/m·K]
\(\rho=\)density [kg/m^{3}]
\(c_p=\)specific heat capacity [J/kg·K]
The grouping of terms \(\rho c_p\) is volumetric heat capacity [J/m^{3}·K]
The SI units used of thermal diffusivity are then [m^{2}/s]. Thermal diffusivity varies from low values of about 0.1 10^{-6} m^{2}/s to high values of about 1000 10^{-6} m^{2}/s. When the thermal conditions around a material changes, a material with a high thermal diffusivity will reach thermal equilibrium faster than a material with a lower thermal diffusivity.
Thermal diffusivities of some common materials used in composite manufacturing are given in the following table:
Material | Thermal Diffusivity [10^{-6} m^{2}/s] |
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Aluminum | 68.9 |
Steel | 14.2 |
Invar | 2.7 |
Carbon-Epoxy Composite | 0.5 |
The earlier described steady-state example can be turned into a transient state conduction case by considering that the same wall is initially at a temperature T_{0.} throughout, and then that at time t = 0 its right-hand wall is suddenly brought to the temperature T_{1}. As time proceeds, the through-thickness temperature of the wall changes as heat flows in or out of the wall until eventually the linear steady-state distribution described in the previous section is achieved.
For this case, the one-dimensional form of the energy equation becomes\[-\frac{dq_x}{dx}=\rho c_p\frac{\partial T}{\partial t}\]
Therefore,
\(\frac{d}{dx}\Biggl(k\frac{dT}{dx}\Biggr)=\rho c_p\frac{\partial T}{\partial t}\)
Which gives\[\alpha\frac{d^2T}{dx^2}=\frac{\partial T}{\partial t}\]
Where \(\alpha\) is the thermal diffusivity in [m^{2}/s].
The rate of temperature change, and consequently the time to reach thermal equilibrium, therefore depends on the thermal diffusivity of the material.
For a composite laminate curing in an autoclave or oven, where the moving air is acting as a convective heat transfer boundary condition, graphical Heisler charts can be used to look up an approximate solution to the transient conductivity equation for the internal temperature profile of the laminate. Modern technology also allows an exact solution to be obtained using a number of available math or finite element software packages ^{[2]}. Alternatively, a simplified closed-form approximation developed by Rasekh et al. ^{[3]} can also be used. For more information regarding using this approximation, please see the paper by Rasekh et al. here.
Related pages
References
- ↑ [Ref] Gaskell, David R. (1992). An Introduction to Transport Phenomena in Materials Engineering. Macmillan Publishing Company. ISBN 0023407204.CS1 maint: uses authors parameter (link) CS1 maint: date and year (link)
- ↑ [Ref] Slesinger, Nathan Avery (2010). Thermal Modeling Validation Techniques for Thermoset Polymer Matrix Composites (Thesis). doi:10.14288/1.0071063.CS1 maint: uses authors parameter (link)
- ↑ [Ref] Rasekh, Ali et al. (2004). Simple Techniques for Thermal Analysis of the Processing of Composite Structures. Society for the Advancement of Material and Process Engineering.CS1 maint: extra punctuation (link) CS1 maint: uses authors parameter (link) CS1 maint: date and year (link)
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