Introduction
In the previous lessons, we considered the various forms of energy transfer: ‘Thermal Conduction’, ‘Convection’ and ‘Radiation’. A typical application of Finite Element Analysis (FEA) is to assess the heat distribution throughout a solid object when the physical properties of the object are known and the various forms of energy transfer between the boundaries of the object and its surroundings can be evaluated.
The ‘Energy Balance Equation’ is simply a mathematical statement of the First Law of Thermodynamics, i.e.: the total energy of an isolated system is conserved. When we consider a solid object, we can then say that the energy entering, leaving, being stored within or being generated within the solid object will sum to zero:
eq.(12)
© Doug Rattray, LCC, UHI
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Heat Transfer through a Solid
For the time being we shall assume that there is no energy generated or stored within the object (i.e., Q = 0).
Consider a heat conducting solid element (dimensions: Δx, Δy and Δz) with heat transfer boundary conditions at its surfaces as illustrated.
The rate of heat flow by thermal conduction into and out of the object is given by ‘qi’ and ‘qo’ respectively and is described by eq. 3:
Therefore, from eq. 12: '' can be expressed as:
eq.(13)
where ‘ΔyΔz’ describes the surface area ‘A’
© Doug Rattray, LCC, UHI
It should be noted that in eq. 13: the two first derivative terms (dT/dx) consider the temperature gradient at the left- and right-hand side surfaces of our element rather than across the centre of our element as we would like. These terms can be defined across the centre of the element as follows:
Left-hand side surface
Right-hand side surface
The next step is to evaluate this first derivative terms using Taylor Series Expansions. We shall do one expansion for and a second for .
Note: the standard Taylor Series Expansion is typically represented as:
As we continue, we are going to neglect all higher order terms above in the above expansion. This implies that the resulting solution approximates the true solution but shall not be exact, i.e., our solution shall contain some degree of error. Using this approximation is fundamental to the concept of discretisation and Finite Element Analysis.
Applying the Taylor Series Expansion to our left- and right-hand side equations:
Left-hand side surface
Right-hand side surface
Remembering that:
Therefore, the temperature gradients at each surface can be expressed as:
And eq. (13) can be rewritten as:
Rearranging terms in this equation we find that it simplifies to:
To complete our analysis of the heat transfer through the element, we repeat the above process for surfaces in the ‘y’ and ‘z’ planes and sum all heat transfers into and out of the element, which gives:
eq.(14)
The Energy Balance Equation (eq. (12)) presented previously tells us that the above formula (describing the total heat transfer into or out of the element) equals the sum of the heat stored in the element and any heat generated within the element.
Heat Stored and Generated within an Element
The energy stored within a solid object as a function of its rate of change of temperature is dependent upon its mass ‘m’, its specific heat capacity ‘Cp’ such that:
Given that the mass of an object can be described in terms of its density ‘ρ’ and volume; the above equation can be written in terms of its dimensions (in our case: ):
eq.(15)
© Doug Rattray, LCC, UHI
The energy generated from within a solid object by any means (be it electrical, chemical or any other process) is given the notation ‘Q’ and stated in terms of energy generated by unit volume. As such, the total energy generated from within our element is given by:
eq.(16)
Noting that the volume of our element ; we can now substitute the energy entering and leaving the solid object, eq. (14), being stored within the object, eq. (15), and being generated within the solid object, eq. (16), back into the Energy Balance Equation, eq. (12):
eq.(17)
Alternative Expressions of the Energy Balance Equation
As a final point to note, students may come across the Energy Balance Equation eq. (17) expressed slightly differently:
If we divide eq. (17) by , the Energy Balance Equation expresses the ‘Volumetric Heat Balance Equation’:
eq.(17)
If there is no heat stored or generated within the solid, the Energy Balance Equation expresses ‘Laplace’s Equation’:
eq.(18)
In general, the Energy Balance Equation should be integrated across its entire volume:
eq.(19)