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A variety of boundary conditions are encountered in heat transfer analysis:
Temperature conditions
The temperature may be prescribed at specific points and surfaces of the body, denoted by S_{\theta} in (7.2).
Heat flow conditions
The heat flow input may be prescribed at specific points and surfaces of the body. These heat flow boundary conditions are specified in (7.3).
Convection boundary conditions
Included in (7.3) are convection boundary conditions where
q ^ {s} = h (\theta_ {e} - \theta^ {s}) \tag {7.4}
and h is the convection coefficient, which may be temperature-dependent. Here the environmental temperature \theta_{e} is known, but the surface temperature \theta^{S} is unknown.
Radiation boundary conditions
Radiation boundary conditions are also specified in (7.3) with
q ^ {s} = \kappa (\theta_ {r} - \theta^ {s}) \tag {7.5}
where \theta_{r} is the known temperature of the external radiative source and \kappa is a coefficient, evaluated using absolute temperatures,
\kappa = h _ {r} [ (\theta_ {r}) ^ {2} + (\theta^ {S}) ^ {2} ] (\theta_ {r} + \theta^ {S}) \tag {7.6}
The variable h_{r} is determined from the Stefan-Boltzmann constant, the emissivity of the radiant and absorbing materials, and the geometric view factors.
We assume here that \theta_{r} is known. If, on the other hand, the situation of two bodies radiating heat to each other is considered, the analysis is considerably more complicated (see Example 7.6 for such a case).
In addition to these boundary conditions the temperature initial conditions must also be specified in a transient analysis.
For the finite element solution of the heat transfer problem we use the principle of virtual temperatures given as
\int_ {V} \overline {{{\boldsymbol {\theta}}}} ^ {\prime T} \mathbf {k} \boldsymbol {\theta} ^ {\prime} d V = \int_ {V} \overline {{{\theta}}} q ^ {B} d V + \int_ {S _ {q}} \overline {{{\theta}}} ^ {S} q ^ {S} d S + \sum_ {i} \overline {{{\theta}}} ^ {i} Q ^ {i} \tag {7.7}
where
\boldsymbol {\theta} ^ {\prime T} = \left[ \begin{array}{c c c} \frac {\partial \theta}{\partial x} & \frac {\partial \theta}{\partial y} & \frac {\partial \theta}{\partial z} \end{array} \right] \tag {7.8}
\mathbf {k} = \left[ \begin{array}{l l l} k _ {x} & 0 & 0 \\ 0 & k _ {y} & 0 \\ 0 & 0 & k _ {z} \end{array} \right] \tag {7.9}
and the Q^{i} are concentrated heat flow inputs. Each Q^{i} is equivalent to a surface heat flow input over a very small area. The bar over the temperature \theta indicates that a virtual temperature distribution is being considered.
The principle of virtual temperatures is an equation of heat flow equilibrium: for \theta to be the solution of the temperature in the body under consideration, (7.7) must hold for arbitrary virtual (continuous) temperature distributions that are zero on S_{\theta} .
We note that the principle of virtual temperatures is an expression like the principle of virtual displacements used in stress analysis (see Section 4.2). We use the principle of virtual temperatures in the same way as the principle of virtual displacements, and indeed all procedures discussed in Chapters 4 and 5 are directly applicable, except that we now only have the scalar of unknown temperature, whereas in the previous discussion we solved for the vector of unknown displacements.
To further deepen our understanding of the principle of virtual temperatures, we derive the expression in (7.7) in the following example (this derivation is analogous to the presentation in Example 4.2).
EXAMPLE 7.1: Derive the principle of virtual temperatures from the basic differential equations (7.1) to (7.3).
Here we follow the procedure in Example 4.2 (see also Section 3.3.4).
Let us write the governing heat transfer equations in indicial notation. Using x_{1} \equiv x , x_{2} \equiv y , x_{3} \equiv z , and the earlier definitions, we obtain the following.
The differential heat flow equilibrium equation to be satisfied throughout the body
(k _ {i} \theta_ {, i}), _ {i} + q ^ {B} = 0 \quad \text { no sum on } i \text { in parentheses } \tag {a}
The essential boundary condition
\theta = \theta^ {s} \quad \text { on } S _ {\theta} \tag {b}
The natural boundary condition
k _ {n} \theta_ {, n} = q ^ {S} \quad \text { on } S _ {q} \tag {c}
where S = S_{\theta} \cup S_{q}, S_{\theta} \cap S_{q} = 0 .
Let us consider any arbitrarily chosen continuous temperature distribution \bar{\theta} , with \bar{\theta} = 0 on S_{\theta} . Then we have
\int_ {V} \left[ \left(k _ {i} \theta_ {, i}\right) _ {, i} + q ^ {B} \right] \bar {\theta} d V = 0 \tag {d}
We call \bar{\theta} the “virtual temperature distribution.” Since \bar{\theta} is arbitrary, (d) can be satisfied if and only if the quantity in the brackets vanishes. Hence, (d) is equivalent to (a).
Our objective is to now transform (d) such that we lower the order of derivatives in the integral (from second to first order), and we can introduce the natural boundary condition (c). For this purpose we use the mathematical identity
[ \bar {\theta} (k _ {i} \theta_ {, i}) ] _ {, i} = \bar {\theta} _ {, i} (k _ {i} \theta_ {, i}) + \bar {\theta} (k _ {i} \theta_ {, i}) _ {, i}
to transform the relation in (d), to obtain
\int_ {V} \left\{\left[ \bar {\theta} (k _ {i} \theta_ {, i}) \right] _ {, i} - \bar {\theta} _ {, i} (k _ {i} \theta_ {, i}) + q ^ {B} \bar {\theta} \right\} d V = 0 \tag {e}
Our objective is now achieved by using the divergence theorem (see also Example 4.2). We have
\int_ {V} \left[ \bar {\theta} (k _ {i} \theta_ {, i}) \right] _ {, i} d V = \int_ {S} \left[ \bar {\theta} (k _ {i} \theta_ {, i}) \right] n _ {i} d S = \int_ {S} \bar {\theta} (k _ {n} \theta_ {, n}) d S
We thus obtain from (e)
\int_ {V} \left[ - \bar {\theta} _ {, i} (k _ {i} \theta_ {, i}) + q ^ {B} \bar {\theta} \right] d V + \int_ {S} \bar {\theta} (k _ {n} \theta_ {, n}) d S = 0
In light of (c) and the condition that \bar{\theta}=0 on S_{\theta} , we therefore have the required result
\int_ {V} \overline {{{\theta}}} _ {, i} (k _ {i} \theta_ {, i}) d V = \int_ {V} \overline {{{\theta}}} q ^ {B} d V + \int_ {S _ {q}} \overline {{{\theta}}} ^ {s} q ^ {s} d S
where we note that the prescribed heat flux condition (the natural boundary condition) now appears as a forcing term on the right-hand side of the equation.
It is also of value to recognize that the principle of virtual temperatures corresponds to the condition of stationarity of the following functional
\Pi = \int_ {v} \frac {1}{2} \left[ k _ {x} \left(\frac {\partial \theta}{\partial x}\right) ^ {2} + k _ {y} \left(\frac {\partial \theta}{\partial y}\right) ^ {2} + k _ {z} \left(\frac {\partial \theta}{\partial z}\right) ^ {2} \right] d V - \int_ {v} \theta q ^ {B} d V - \int_ {s _ {q}} \theta^ {S} q ^ {S} d S - \sum_ {i} \theta^ {i} Q ^ {i} \tag {7.10}
Namely, invoking \delta \Pi = 0 , we obtain
\int_ {V} \delta \boldsymbol {\theta} ^ {\prime T} \mathbf {k} \boldsymbol {\theta} ^ {\prime} d V = \int_ {V} \delta \theta q ^ {B} d V + \int_ {S _ {q}} \delta \theta^ {S} q ^ {S} d S + \sum_ {i} \delta \theta^ {i} Q ^ {i} \tag {7.11}
where \delta\theta can be arbitrary but must be zero on S_{\theta} . Using integration by parts (i.e., the divergence theorem) on (7.11) we can of course extract the governing differential equation of equilibrium (7.1) and the heat flow boundary condition (7.3) (which in essence corresponds to reversing the process used in Example 7.1; see Example 3.18). However, on comparing (7.11) with (7.7), we recognize that (7.11) is the principle of virtual temperatures with \delta\theta \equiv \overline{\theta} .
In the heat transfer problem considered above, we assumed steady-state conditions. However, when significant heat flow input changes are specified over a “short” time period (due to a change of any of the boundary conditions or the heat generation in the body), this period being short measured on the natural time constants of the system (given by the thermal eigenvalues; see Chapter 9), it is important to include a term that takes account of the rate at which heat is stored within the material. This rate of heat absorption is
q ^ {c} = \rho c \dot {\theta} \tag {7.12}
where c is the material specific heat capacity. The variable q^{c} can be understood to be part of the heat generated—of course, q^{c} must be subtracted from the otherwise generated heat q^{B} in (7.7) because it is heat stored—and the effect leads to a transient response solution.
7.2.2 Incremental Equations
The principle of virtual temperatures expresses the heat flow equilibrium at all times of interest. For a general solution scheme of both linear and nonlinear, steady-state and
transient problems we aim to develop incremental equilibrium equations. As in an incremental finite element stress analysis (see Section 6.1), assume that the conditions at time t have been calculated and that the temperatures are to be determined for time t + \Delta t , where \Delta t is the time increment.
Steady-State Conditions
Considering first steady-state conditions, in which the time stepping is merely used to describe the heat flow loading, the principle of virtual temperatures applied at time t + \Delta t gives
\begin{array}{l} \int_ {V} \overline {{{\boldsymbol {\theta}}}} ^ {\prime T} t + \Delta t \mathbf {k} ^ {\prime + \Delta t} \boldsymbol {\theta} ^ {\prime} d V \\ = ^ {t + \Delta t} \mathcal {Q} + \int_ {S _ {c}} \bar {\theta} ^ {S} {} ^ {t + \Delta t} h \left(^ {t + \Delta t} \theta_ {e} - ^ {t + \Delta t} \theta^ {S}\right) d S + \int_ {S _ {r}} \bar {\theta} ^ {S} {} ^ {t + \Delta t} \kappa \left(^ {t + \Delta t} \theta_ {r} - ^ {t + \Delta t} \theta^ {S}\right) d S \tag {7.13} \\ \end{array}
where the superscript t + \Delta t denotes “at time t + \Delta t ,” S_c and S_r are the surface areas with convection and radiation boundary conditions, respectively, and ^{t+\Delta t} \mathfrak{Q} corresponds to further external heat flow input to the system at time t + \Delta t . Note that in (7.13) the temperatures ^{t+\Delta t} \theta_e and ^{t+\Delta t} \theta_r are known, whereas ^{t+\Delta t} \theta^S is the unknown surface temperature on S_c and S_r . The quantity ^{t+\Delta t} \mathfrak{Q} includes the effects of the internal heat generation ^{t+\Delta t} q^B , the surface heat flux inputs ^{t+\Delta t} q^S that are not included in the convection and radiation boundary conditions, and the concentrated heat flow inputs ^{t+\Delta t} Q^i ,
{ } ^ { t + \Delta t } \mathfrak { Q } = \int _ { v } \bar { \theta } { } ^ { t + \Delta t } q ^ { B } d V + \int _ { s _ { q } } \bar { \theta } ^ { S } { } ^ { t + \Delta t } q ^ { S } d S + \sum _ { i } \bar { \theta } ^ { i } { } ^ { t + \Delta t } Q ^ { i } \tag {7.14}
Considering the general heat flow equilibrium relation in (7.13), we note that in linear analysis {}^{t+\Delta t}k and {}^{t+\Delta t}h are constant and radiation boundary conditions are not included. Hence, the relation in (7.13) can be rearranged to obtain in linear analysis,
\int_ {V} \overline {{{\theta}}} ^ {\prime T} \mathbf {k} ^ {\prime + \Delta t} \theta^ {\prime} d V + \int_ {S _ {c}} \overline {{{\theta}}} ^ {S} h ^ {\prime + \Delta t} \theta^ {S} d S = ^ {\prime + \Delta t} 2 + \int_ {S _ {c}} \overline {{{\theta}}} ^ {S} h ^ {\prime + \Delta t} \theta_ {e} d S \tag {7.15}
and it is possible to solve directly for the unknown temperature t^{+\Delta t}\theta .
In general nonlinear heat transfer analysis the relation in (7.13) is a nonlinear equation in the unknown temperature at time t + \Delta t . An approximate solution for this temperature can be obtained by incrementally decomposing (7.13) as summarized in Table 7.1. As in stress analysis (see Section 6.1), this decomposition can be understood to be the first step of a Newton-Raphson iteration for heat flow equilibrium in which
{ } ^ { t + \Delta t } \theta ^ { ( i ) } = { } ^ { t + \Delta t } \theta ^ { ( i - 1 ) } + \Delta \theta ^ { ( i ) } \tag {7.16}
where ^{t+\Delta t}\theta^{(i-1)} is the temperature distribution at the end of iteration (i-1) and \Delta\theta^{(i)} is the temperature increment in iteration (i) ; also, ^{t+\Delta t}\theta^{(0)} = ^{t}\theta . In Table 7.1 we use \theta to describe \Delta\theta^{(1)} and consider the equation for the first iteration.
In a full Newton-Raphson iteration the accurate solution of (7.13) would be obtained by using (7.16) and updating all variables in the incremental equation of Table 7.1. in each
TABLE 7.1 Incremental nonlinear heat flow equilibrium equation
- Equilibrium equation at time
t + \Delta t
\int_ {V} \overline {{{\theta}}} ^ {\prime T} {} ^ {t + \Delta t} \mathbf {k} ^ {t + \Delta t} \theta^ {t} d V = ^ {t + \Delta t} \mathfrak {Q} + \int_ {s _ {c}} \overline {{{\theta}}} ^ {S} {} ^ {t + \Delta t} h ^ {(t + \Delta t} \theta_ {e} - ^ {t + \Delta t} \theta^ {S)} d S + \int_ {s _ {r}} \overline {{{\theta}}} ^ {S} {} ^ {t + \Delta t} \kappa^ {(t + \Delta t} \theta_ {r} - ^ {t + \Delta t} \theta^ {S)} d S
- Linearization of equation
We use: t^{+\Delta t}\theta = {}^{\prime}\theta +\theta ;{}^{t + \Delta t}\theta^{\prime} = {}^{\prime}\theta^{\prime} + \theta^{\prime}; \tilde{\kappa} = 4^{\prime}h_{r}(^{\prime}\theta^{S})^{3}
^ \prime \kappa = ^ {\prime} h _ {r} ((^ {\prime + \Delta t} \theta_ {r}) ^ {2} + (^ {\prime} \theta^ {S}) ^ {2}) (^ {\prime + \Delta t} \theta_ {r} + ^ {\prime} \theta^ {S})
Substituting into the equation of heat flow equilibrium, we obtain
\begin{array}{l} \int_ {V} \bar {\theta} ^ {\prime T} {} ^ {\prime} \mathbf {k} \theta^ {\prime} d V + \int_ {S _ {c}} \bar {\theta} ^ {S} {} ^ {\prime} h \theta^ {S} d S + \int_ {S _ {r}} \bar {\theta} ^ {S} {} ^ {\prime} \tilde {\kappa} \theta^ {S} d S = ^ {\prime + \Delta t} \mathfrak {Q} + \int_ {S _ {c}} \bar {\theta} ^ {S} {} ^ {\prime} h \left(^ {\prime + \Delta t} \theta_ {e} - ^ {\prime} \theta^ {S}\right) d S \\ + \int_ {S _ {r}} \overline {{\theta}} ^ {s} {} ^ {\prime} \kappa \left(^ {t + \Delta t} \theta_ {r} - ^ {\prime} \theta^ {s}\right) d S - \int_ {V} \overline {{\theta}} ^ {\prime T} {} ^ {\prime} \mathbf {k} ^ {\prime} \theta^ {\prime} d V \\ \end{array}
iteration. Hence, we solve for i = 1, 2, \ldots ,
\begin{array}{l} \int_ {V} \overline {{{\theta}}} ^ {\prime T} {} ^ {t + \Delta t} \mathbf {k} ^ {(i - 1)} \Delta \theta^ {\prime (i)} d V + \int_ {S _ {c}} \overline {{{\theta}}} ^ {S} {} ^ {t + \Delta t} h ^ {(i - 1)} \Delta \theta^ {S (i)} d S + \int_ {S _ {r}} \overline {{{\theta}}} ^ {S} {} ^ {t + \Delta t} \tilde {\kappa} ^ {(i - 1)} \Delta \theta^ {S (i)} d S \\ = ^ {t + \Delta t} 2 + \int_ {S _ {c}} \bar {\theta} ^ {S t + \Delta t} h ^ {(i - 1)} \left(^ {t + \Delta t} \theta_ {e} - ^ {t + \Delta t} \theta^ {S (i - 1)}\right) d S \tag {7.17} \\ + \int_ {S _ {r}} \overline {{\theta}} ^ {S} {} ^ {t + \Delta t} \kappa^ {(i - 1)} \left(^ {t + \Delta t} \theta_ {r} - ^ {t + \Delta t} \theta^ {S (i - 1)}\right) d S - \int_ {V} \overline {{\boldsymbol {\theta}}} ^ {\prime T} {} ^ {t + \Delta t} \mathbf {k} ^ {(i - 1)} {} ^ {t + \Delta t} \boldsymbol {\theta} ^ {\prime (i - 1)} d V \\ \end{array}
where t+\Delta t h^{(i-1)} , t+\Delta t \kappa^{(i-1)} , and t+\Delta t \mathbf{k}^{(i-1)} are the convection and radiation coefficients and the conductivity constitutive matrix that correspond to the temperature t+\Delta t \theta^{(i-1)} .
Frequently, in practice, the modified Newton-Raphson iteration is employed, in which case the left-hand side of (7.17) is evaluated only at the beginning of the time step and not updated until the next time increment (see Section 8.4.1).
Although it might appear that an actual linearization of the heat flow equilibrium equation is achieved in Table 7.1, a closer study shows that the equations in the table correspond to only an approximate linearization. Consequently, (7.17) is, in general, also not a full linearization about the state of the last iteration. The difficulty lies in that the tangent relations of the material constants, that is, of the conduction, convection, and radiation coefficients when temperature-dependent, need to be included in the linearization, and this can be achieved only when the functional relationship between the material property and temperature is given in analytical form. We demonstrate this observation in the following example.
EXAMPLE 7.2: Consider the analysis of the slab shown in Fig. E7.2. Establish the incremental form of the principle of virtual temperatures for the modified Newton-Raphson iteration and for the full Newton-Raphson iteration.
text_image
Uniform convection with coefficient h; h = 2 + θ Uniform radiation with hr = constant Temperature θe(t) = 20°C θr(t) = 100°C Conductivity k = 10 + 2θ Uniform heat flow qs 2 3 1 r = -1 r = 0 r = +1 x L ∞
Figure E7.2 Analysis of an infinite slab
The principle of virtual temperatures for the one-dimensional problem, considering a unit cross-sectional area of the slab, is
\begin{array}{l} \int_ {0} ^ {L} \overline {{{\theta}}} ^ {\prime} {} ^ {t + \Delta t} k ^ {t + \Delta t} \theta^ {\prime} d x \tag {a} \\ = \left. \left[ \bar {\theta} ^ {S} q ^ {S} \right] \right| _ {x = 0} + \left. \left[ \bar {\theta} ^ {S t + \Delta t} h \left(^ {t + \Delta t} \theta_ {e} - ^ {t + \Delta t} \theta^ {S}\right) \right] \right| _ {x = L} + \left. \left[ \bar {\theta} ^ {S t + \Delta t} \kappa \left(^ {t + \Delta t} \theta_ {r} - ^ {t + \Delta t} \theta^ {S}\right) \right] \right| _ {x = L} \\ \end{array}
where ^{t+\Delta t}\theta' = \partial^{t+\Delta t}\theta/\partial x , \overline{\theta}' = \partial\overline{\theta}/\partial x , and ^{t+\Delta t}\kappa is evaluated using degrees Kelvin.
The incremental form in the modified Newton-Raphson iteration is based on the decomposition given (for the first iteration) in Table 7.1,
\begin{array}{l} \int_ {0} ^ {L} \bar {\theta} ^ {\prime} (1 0 + 2 ^ {\prime} \theta) \Delta \theta^ {\prime (i)} d x + [ \bar {\theta} ^ {S} (2 + ^ {\prime} \theta^ {S}) \Delta \theta^ {S (i)} ] | _ {x = L} + [ \bar {\theta} ^ {S} 4 h _ {r} (^ {\prime} \theta^ {S}) ^ {3} \Delta \theta^ {S (i)} ] | _ {x = L} \\ = \left[ \bar {\theta} ^ {S} q ^ {S} \right] \big | _ {x = 0} + \left[ \bar {\theta} ^ {S} \left(2 + ^ {t + \Delta t} \theta^ {S (i - 1)}\right) \left(2 0 - ^ {t + \Delta t} \theta^ {S (i - 1)}\right) \right] \big | _ {x = L} \tag {b} \\ + \left. \{\bar {\theta} ^ {S t + \Delta t} \kappa^ {(i - 1)} [ 1 0 0 - ^ {t + \Delta t} \theta^ {S (i - 1)} ] \} \right| _ {x = L} - \int_ {0} ^ {L} \bar {\theta} ^ {\prime} (1 0 + 2 ^ {t + \Delta t} \theta^ {(i - 1)}) ^ {t + \Delta t} \theta^ {\prime (i - 1)} d x \\ \end{array}
In the full Newton-Raphson iteration the same right-hand side is used, but the left-hand side is given by
\begin{array}{l} \text { Left - hand side } = \int_ {0} ^ {L} \overline {{\theta}} ^ {\prime} (1 0 + 2 ^ {t + \Delta t} \theta^ {(i - 1)}) \Delta \theta^ {\prime (i)} d x + [ \overline {{\theta}} ^ {S} (2 + ^ {t + \Delta t} \theta^ {S (i - 1)}) \Delta \theta^ {S (i)} ] | _ {x = L} \\ + \left[ \bar {\theta} ^ {S} 4 h _ {r} \left(^ {t + \Delta t} \theta^ {S (i - 1)}\right) ^ {3} \Delta \theta^ {S (i)} \right] | _ {x = L} \tag {c} \\ \end{array}
The actual linearization, however, is obtained by differentiating the equation of the principle of virtual temperatures about the last calculated state and using the analytical expressions.
Let us consider as an example the conduction term in (a) and use the procedure of linearization about the state at time t that we employed in Section 6.3.1. Hence, we use the Taylor series expansion
\bar {\theta} ^ {\prime} {} ^ {\prime + \Delta} q \doteq \bar {\theta} ^ {\prime} {} ^ {\prime} q + \frac {\partial}{\partial \theta} (\bar {\theta} ^ {\prime} {} ^ {\prime} q) d \theta
However, since (\partial \overline{\theta}' / \partial \theta) = 0 , we have
\frac {\partial}{\partial \theta} (\overline {{{\theta}}} ^ {\prime} {} ^ {\prime} q) d \theta = \left[ \overline {{{\theta}}} ^ {\prime} \frac {\partial}{\partial \theta} (^ {\prime} q) \right] d \theta
Now substituting for ^t q = -(10 + 2^t \theta)(\partial^t \theta / \partial x) , we obtain
\overline {{{\theta}}} ^ {\prime} {} ^ {\prime + \Delta t} q \doteq - \overline {{{\theta}}} ^ {\prime} \left(\underbrace {[ 1 0 + 2 ^ {\prime} \theta ] \frac {\partial^ {\prime} \theta}{\partial x} + 2 \frac {\partial^ {\prime} \theta}{\partial x} d \theta} _ {\text { Term 1 }} + \underbrace {(1 0 + 2 ^ {\prime} \theta) d \theta^ {\prime}} _ {\text { Term 2 }}\right) \tag {d}
We note that the term 1 and term 3 on the right-hand side of (d) are included in the incremental principle of virtual temperatures given in (b) [and in (c)] but that term 2 is an extra expression not accounted for in (b), (c), and Table 7.1. In the finite element solution to the slab, this term would lead to a nonsymmetric tangent conductivity matrix.
Also, in a similar manner, the actual linearization of the convection and radiation terms can be obtained. This development shows that for the convection part a temperature-dependent term is also neglected, whereas the linearization of the radiation part is complete because in this example the h_{r} -coefficient is temperature-independent (see Exercise 7.2).
As shown above in a specific example, (7.17) does not, in general, correspond to the exact linearization of the principle of virtual temperatures about the last calculated temperature state. However, (7.17) represents a general iterative solution scheme which, in particular, can be applied when the material relationships are given piecewise linear as a function of temperature (such a definition can be convenient in the use of a general program implementation that is not based on specific analytical expressions of material properties). If iteration convergence is obtained, the correct solution of the principle of virtual temperatures (7.7) has been calculated [since the equation (7.7) is satisfied when the right-hand side in (7.17) is zero], and frequently in practice only a few iterations are needed for reasonable time (load) step magnitudes.
Of course, if specific analytical relationships of the material constants are to be used and convergence difficulties are encountered with (7.17), it may be advantageous to use the exact linearization of the principle of virtual temperatures in the iterative solution (see Exercise 7.3).
Transient Conditions
In transient analysis, the heat capacity effect is included in much the same way as we introduced the inertia forces in stress analysis (see Sections 4.2.1 and 6.2.3).
The principle of virtual temperatures at time t + \Delta t is now
\begin{array}{l} \int_ {V} \overline {{{\theta}}} ^ {T} {} ^ {t + \Delta t} (\rho c) ^ {t + \Delta t} \dot {\theta} d V + \int_ {V} \overline {{{\theta}}} ^ {\prime T} {} ^ {t + \Delta t} \mathbf {k} ^ {t + \Delta t} \boldsymbol {\theta} ^ {\prime} d V \\ = ^ {t + \Delta t} \mathfrak {Q} + \int_ {S _ {c}} \bar {\theta} ^ {S t + \Delta t} h \left(^ {t + \Delta t} \theta_ {e} - ^ {t + \Delta t} \theta^ {S}\right) d S + \int_ {S _ {r}} \bar {\theta} ^ {S t + \Delta t} \kappa \left(^ {t + \Delta t} \theta_ {r} - ^ {t + \Delta t} \theta^ {S}\right) d S \tag {7.18} \\ \end{array}
where t^{+\Delta t}\Omega is defined as in (7.14), but t^{+\Delta t}q^{B} is now the rate of heat generation excluding the heat capacity effect.
The relation in (7.18) is used to calculate the temperature at time t + \Delta t when an implicit time integration method is employed (such as the Euler backward method). On the other hand, in an explicit time integration scheme, the principle of virtual temperatures is applied at time t to calculate the unknown temperature at time t + \Delta t (see Sections 7.2.3 and 9.6). Whereas a Newton-Raphson iterative method including the heat capacity effects is used in implicit integration (when nonlinearities are present), a simple forward integration without iteration is employed with an explicit method.
7.2.3 Finite Element Discretization of Heat Transfer Equations
The finite element solution of the heat transfer governing equations is obtained using procedures analogous to those employed in stress analysis. We consider first the analysis of steady-state conditions. Assume that the complete body under consideration has been idealized as an assemblage of finite elements; then, in analogy to stress analysis we have at time t + \Delta t for element m,
\mathbf {t} ^ {+ \Delta t} \theta^ {(m)} = \mathbf {H} ^ {(m)} \mathbf {t} ^ {+ \Delta t} \theta
{ } ^ { t + \Delta t } \theta ^ { S ( m ) } = \mathbf { H } ^ { S ( m ) t + \Delta t } \boldsymbol { \theta } \tag {7.19}
{ } ^ { t + \Delta t } \boldsymbol { \theta } ^ { \prime ( m ) } = \mathbf { B } ^ { ( m ) } { } ^ { t + \Delta t } \boldsymbol { \theta }
where the superscript (m) denotes element m and t+\Delta t\theta is a vector of all nodal point temperatures at time t+\Delta t ,
{ } ^ { t + \Delta t } \boldsymbol { \theta } ^ { T } = \left[ { } ^ { t + \Delta t } \theta _ { 1 } { } ^ { t + \Delta t } \theta _ { 2 } \dots { } ^ { t + \Delta t } \theta _ { n } \right] \tag {7.20}
The matrices \mathbf{H}^{(m)} and \mathbf{B}^{(m)} are the element temperature and temperature-gradient interpolation matrices, respectively, and the matrix \mathbf{H}^{S(m)} is the surface temperature interpolation matrix. We evaluate in (7.19) the element temperatures and temperature gradients at time t + \Delta t , but the same interpolation matrices are also employed to calculate the element temperature conditions at any other time, and hence for incremental temperatures and incremental temperature gradients.
Linear Steady-State Conditions
Using the relations in (7.19) and substituting into (7.15), we obtain the finite element governing equations in linear heat transfer analysis:
\left(\mathbf {K} ^ {k} + \mathbf {K} ^ {c}\right) ^ {t + \Delta t} \boldsymbol {\theta} = ^ {t + \Delta t} \mathbf {Q} + ^ {t + \Delta t} \mathbf {Q} ^ {e} \tag {7.21}
where K^{k} is the conductivity matrix,
\mathbf {K} ^ {k} = \sum_ {m} \int_ {V ^ {(m)}} \mathbf {B} ^ {(m) ^ {T}} \mathbf {k} ^ {(m)} \mathbf {B} ^ {(m)} d V ^ {(m)} \tag {7.22}
and \mathbf{K}^c is the convection matrix,
\mathbf {K} ^ {c} = \sum_ {m} \int_ {S _ {c} ^ {(m)}} h ^ {(m)} \mathbf {H} ^ {S (m) ^ {T}} \mathbf {H} ^ {S (m)} d S ^ {(m)} \tag {7.23}
The nodal point heat flow input vector t+\Delta tQ is given by
{ } ^ { t + \Delta t } \mathbf { Q } = { } ^ { t + \Delta t } \mathbf { Q } _ { B } + { } ^ { t + \Delta t } \mathbf { Q } _ { S } + { } ^ { t + \Delta t } \mathbf { Q } _ { C } \tag {7.24}
where t + \Delta t\mathbf{Q}_B = \sum_m\int_{V(m)}\mathbf{H}^{(m)T}{}_{t + \Delta t}q^{B(m)}dV^{(m)} (7.25)
{ } ^ { t + \Delta t } \mathbf { Q } _ { S } = \sum _ { m } \int _ { S _ { q } ^ { ( m ) } } \mathbf { H } ^ { S ( m ) ^ { T } t + \Delta t } q ^ { S ( m ) } d S ^ { ( m ) } \tag {7.26}
and t^{+\Delta t} \mathbf{Q}_c is a vector of concentrated nodal point heat flow input. The nodal point heat flow contribution t^{+\Delta t} \mathbf{Q}^e is due to the convection boundary conditions. Using the element surface temperature interpolations to define the environmental temperature t^{+\Delta t} \theta_e on the element surfaces in terms of the given nodal point environmental temperatures t^{+\Delta t} \theta_e , we have
{ } ^ { t + \Delta t } \mathbf { Q } ^ { e } = \sum _ { m } \int _ { S _ { c } ^ { ( m ) } } h ^ { ( m ) } \mathbf { H } ^ { S ( m ) ^ { T } } \mathbf { H } ^ { S ( m ) } { } ^ { t + \Delta t } \boldsymbol { \theta } _ { e } d S ^ { ( m ) } \tag {7.27}
The above formulation is effectively used with the variable-number-nodes isoparametric finite elements discussed in Chapter 5. We demonstrate the calculation of the element matrices in the following example.
EXAMPLE 7.3: Consider the four-node isoparametric element in Fig. E7.3. Discuss the calculation of the conductivity matrix \mathbf{K}^k , convection matrix \mathbf{K}^c , and heat flow input vectors ^{r + \Delta t}\mathbf{Q}_B and ^{r + \Delta t}\mathbf{Q}^e .
For the evaluation of these matrices we need the matrices H, B, H^{s} , and k. The temperature interpolation matrix H is composed of the interpolation functions defined in Fig. 5.4,
\mathbf {H} = \frac {1}{4} [ (1 + r) (1 + s) \quad (1 - r) (1 + s) \quad (1 - r) (1 - s) \quad (1 + r) (1 - s) ]
We obtain \mathbf{H}^s by evaluating \mathbf{H} at r = 1 , so that
\mathbf {H} ^ {s} = \frac {1}{2} [ (1 + s) \quad 0 \quad 0 \quad (1 - s) ]
To evaluate B we first evaluate the Jacobian operator J (see Example 5.3):
\mathbf {J} = \left[ \begin{array}{l l} 1 & \frac {1 + s}{4} \\ 0 & \frac {3 + r}{4} \end{array} \right]
Hence
\mathbf {B} = \frac {1}{4} \left[ \begin{array}{c c} 1 & - \left(\frac {1 + s}{3 + r}\right) \\ 0 & \frac {4}{(3 + r)} \end{array} \right] \left[ \begin{array}{c c c c} (1 + s) & - (1 + s) & - (1 - s) & (1 - s) \\ (1 + r) & (1 - r) & - (1 - r) & - (1 + r) \end{array} \right]
= \frac {1}{4 (3 + r)} \left[ \begin{array}{c c c c} 2 (1 + s) & - 4 (1 + s) & 2 (2 s - r - 1) & 2 (2 + r - s) \\ 4 (1 + r) & 4 (1 - r) & - 4 (1 - r) & - 4 (1 + r) \end{array} \right]
Finally, we have \mathbf{k} = \begin{bmatrix} k & 0 \\ 0 & k \end{bmatrix}
text_image
Thickness = t Conductivity = k s 1 2 cm r 2 cm 3 4 2 cm Convective boundary condition with constant h y x
Figure E7.3 Four-node element in heat transfer conditions
The element matrices can now be evaluated using numerical integration as in the analysis of solids and structures (see Chapter 5).
Nonlinear Steady-State Conditions
For general nonlinear analysis the temperature and temperature gradient interpolations of (7.19) are substituted into the heat flow equilibrium relation (7.17) to obtain
\left(^ {t + \Delta t} \mathbf {K} ^ {k (i - 1)} + ^ {t + \Delta t} \mathbf {K} ^ {c (i - 1)} + ^ {t + \Delta t} \mathbf {K} ^ {r (i - 1)}\right) \Delta \boldsymbol {\theta} ^ {(i)} = ^ {t + \Delta t} \mathbf {Q} + ^ {t + \Delta t} \mathbf {Q} ^ {c (i - 1)} + ^ {t + \Delta t} \mathbf {Q} ^ {r (i - 1)} - ^ {t + \Delta t} \mathbf {Q} ^ {k (i - 1)} \tag {7.28}
where the nodal point temperatures at the end of iteration (i) are
{ } ^ { t + \Delta t } \boldsymbol { \theta } ^ { ( i ) } = { } ^ { t + \Delta t } \boldsymbol { \theta } ^ { ( i - 1 ) } + \Delta \boldsymbol { \theta } ^ { ( i ) } \tag {7.29}
The matrices and vectors used in (7.28) are directly obtained from the individual terms used in (7.17) and are defined in Table 7.2. The nodal point heat flow input vector t^{+\Delta t}Q was already defined in (7.24).
TABLE 7.2 Finite element matrices in nonlinear heat transfer analysis
| Integral | Finite element evaluation |
| $\int_{V} \overline{\theta}^{tT} t + \Delta t \mathbf{k}^{(i-1)} \Delta \theta^{t(i)} dV$ | $^{t+\Delta t} \mathbf{K}^{k(i-1)} \Delta \theta^{(i)} = \left( \sum_{m} \int_{V^{(m)}} \mathbf{B}^{(m)^T} t + \Delta t \mathbf{k}^{(m)(i-1)} \mathbf{B}^{(m)} dV^{(m)} \right) \Delta \theta^{(i)}$ |
| $\int_{S_c} \overline{\theta}^S t + \Delta t h^{(i-1)} \Delta \theta^{S(i)} dS$ | $^{t+\Delta t} \mathbf{K}^{c(i-1)} \Delta \theta^{(i)} = \left( \sum_{m} \int_{S_c^{(m)}} ^{t+\Delta t} h^{(m)(i-1)} \mathbf{H}^{S(m)^T} \mathbf{H}^{S(m)} dS^{(m)} \right) \Delta \theta^{(i)}$ |
| $\int_{S_r} \overline{\theta}^S t + \Delta t \tilde{\kappa}^{(i-1)} \Delta \theta^{S(i)} dS$ | $^{t+\Delta t} \mathbf{K}^{r(i-1)} \Delta \theta^{(i)} = \left( \sum_{m} \int_{S_r^{(m)}} ^{t+\Delta t} \tilde{\kappa}^{(m)(i-1)} \mathbf{H}^{S(m)^T} \mathbf{H}^{S(m)} dS^{(m)} \right) \Delta \theta^{(i)}$ |
| $\int_{S_c} \overline{\theta}^S t + \Delta t h^{(i-1)} (t + \Delta t \theta_e - ^{t+\Delta t} \theta^{S(i-1)}) dS$ | $^{t+\Delta t} \mathbf{Q}^{c(i-1)} = \sum_{m} \int_{S_c^{(m)}} ^{t+\Delta t} h^{(m)(i-1)} \mathbf{H}^{S(m)^T} \left[ \mathbf{H}^{S(m)} (t + \Delta t \theta_e - ^{t+\Delta t} \theta^{(i-1)}) \right] dS^{(m)}$ |
| $\int_{S_r} \overline{\theta}^S t + \Delta t \kappa^{(i-1)} (t + \Delta t \theta_r - ^{t+\Delta t} \theta^{S(i-1)}) dS$ | $^{t+\Delta t} \mathbf{Q}^{r(i-1)} = \sum_{m} \int_{S_r^{(m)}} ^{t+\Delta t} \kappa^{(m)(i-1)} \mathbf{H}^{S(m)^T} \left[ \mathbf{H}^{S(m)} (t + \Delta t \theta_r - ^{t+\Delta t} \theta^{(i-1)}) \right] dS^{(m)}$ |
| $\int_{V} \overline{\theta}^{tT} t + \Delta t \mathbf{k}^{(i-1)} t + \Delta t \theta^{t(i-1)} dV$ | $^{t+\Delta t} \mathbf{Q}^{k(i-1)} = \sum_{m} \int_{V^{(m)}} \mathbf{B}^{(m)^T} \left[ t + \Delta t \mathbf{k}^{(m)(i-1)} \mathbf{B}^{(m)} t + \Delta t \theta^{(i-1)} \right] dV^{(m)}$ |