Quasi-one-dimensional treatment of heat transfer in beams of non-uniform cross-section

We test performance of a quasi-one-dimensional approach to steady state heat transfer in broadening beams, characterized by variable cross-sectional area. The results obtained analytically and by solving the governing equation numerically in Python, for three different instances, are compared to FEM simulations. We demonstrate that the quasi-one-dimensional model gives accurate predictions for beam suitably averaged temperatures but the range of Biot numbers for which it can be accurately applied to is currently limited by a crucial parameter.


Introduction
In our former work we have developed a quasi-one-dimensional approach for the description of heat transfer, where instead of the full temperature field ϑ(x, y, z) one may work with a suitably averaged temperature  ̅ (x) that depends on just one variable of interest.We have applied this approach to study the temperature rise in transformer's windings [1].Later we have derived more general formulation of this approach that uses curved coordinates, the publication of which is now in preparation.One of the outcomes of that work is a specific form of an effective equation for suitably averaged temperature  ̅ (r) in a uniformly broadening beam that will be stated in the next section.We present its analytical and numerical solution and then proceed with their comparison with the exact FEM (finite element method) simulation of a 2D model of the broadening beam.

Model of broadening beam
We consider a symmetric beam of uniformly increasing cross-section that is geometrically given by an annular sector of finite thickness, figure 1a.The two surfaces along the beam's radial length exhibit convective heat transfer with environment defined by heat transfer coefficient α and fluid temperature ϑ∞.On the other hand, the curved surfaces are characterized by prescribed temperatures ϑ0, ϑ1.We consider the heat transfer in the beam as two-dimensional, i.e., there is no heat flow along the third perpendicular coordinate and the temperature along that direction remains constant.The material of the beam is characterized by constant thermal conductivity λ.This kind of geometry and heat transfer has been considered by many authors; several examples can be found in [2][3][4][5][6].The broadening beam model can be parametrized using polar coordinates.Using these the quasi-one-dimensional equation takes the form where ̇ is the heat generation per unit volume and ∆φ is the angle of the annular sector.The variable r attains values from r0 for the first non-convective boundary surface up to the value r1 for the second non-convective boundary surface where ∆r = r1 − r0, figure 1b Figure 1.A a) The 2D geometry of the broadening beam is given by an annular sector.b) The annular sector given by the angle ∆φ and radius r.
Equation 1 features the beam surface temperature at the convective boundaries ϑs(r) and an averaged temperature dependent on r that is averaged over coordinate φ and is defined as where ∆φ = φ1 − φ0.To solve the equation, we introduce an important parameter k, the proportionality factor between the surface and angle-averaged temperature rises above the temperature of the fluid: which changes equation 1 to For the parameter k(r) we use an extension of the formula for a rectangular slab [1] that locally accounts for the broadening of the beam with increasing variable r where h = r∆φ.The formula for the parameter k(r) was derived for the case of a rectangular slab with uniform internal heat generation and zero flux condition at the non-convective boundaries That is why we firstly solve equation 4 for the beam with zero flux conditions at the non-connective boundaries, figure 2b.We get a very complex analytical solution, so instead of using it we proceed to solve equation 4 numerically in Python.Secondly, we solve equation 4 for the initially stated Dirichlet conditions ϑ0, ϑ1, figure 1a.To obtain a relatively simple analytical solution we change parameter k(r) from a variable to a constant by introducing a weighted average value By the same token we introduce a weighted average temperature For no heat generation in the beam ̇ = 0, k as a constant, and non-convective boundaries kept at ϑ0 and ϑ1, equation 4 has a solution where I0 and K0 are modified Bessel functions of first and second kind of zero order respectively, A = (2α ̅ )/(λ∆φ), c1 and c2 are constant to be determined from the boundary conditions, temperatures ϑ0 and ϑ1.

Comparison with FEM simulation
Here ℎ ̅ is the beam's mean width, figure 3. The Biot number characteristic length is one half of the mean width as heat can travel in the direction to either of the convective boundaries.This choice is similar to the one used in [7].From the FEM-obtained two-dimensional temperature field (, ) we extract the angle-averaged temperature  ̅ (rn) at nine equally spread radial positions rn = 1.0 + 0.1n, n = 1, ..., 9 and compare them to the model values obtained from running the equation 4 Python numerical solver (in the case of the beams with internal heat generation that have zero flux non-convective boundaries) and equation 8 (in the case of the beams with no internal generation that have nonconvective boundaries kept at two distinct temperatures).We further compare the values for k(r) at the nine positions obtained from equation 5 with the k(r) values obtained from the FEM simulations.FEM values for k(r) are calculated using the already extracted nine FEM values for the angle-averaged temperature  ̅ (rn) and extracting the nine FEM values for the surface temperature ϑs(rn) and putting them into equation 3. It is important to see, how the parameter k(r), which is derived for one specific instance -a uniformly heated rectangular slab, figure 2a, fares in situations it is specifically designed for (beams with heat generation that have zero flux non-convective boundaries) and situations it is not specifically designed for (beams with no heat generation that have two non-convective boundaries kept at two distinct temperatures).To make quantitative comparison for the quality of the quasi-1D model we calculate mean relative error for  ̅ (r) and k(r): We also compare the FEM and model beam average temperature  ̅ through relative error σ  ̅ .The FEM average temperature is obtained from the simulation.The model average temperature for the beams with internal heat generation that have zero flux non-convective boundaries is calculated by averaging the nine values obtained from the Python solver.The average temperature for the beams with no heat internal heat generation that have two non-convective boundaries kept at two distinct temperatures is obtained from equation 7. Without loss of generality, we set the fluid temperature ϑ∞ to zero and hence the resulting temperatures represent the temperature rises above the fluid temperature.
We will adhere to this convention for the rest of our paper.The values computed for Bi = 100 and Bi = 10 in table 2 marked by an asterix carry larger numerical uncertainty.In these two cases, concerning only the instance with the angle ∆φc, we could determine the constant c1 in equation 8 only approximately.Table 2 shows that as the Biot number decreases, the discrepancies between the FEM and model values for the averaged temperatures correspondingly decrease.The results presented in the table pertain to three instances that differ solely in the angle ∆φ and thermal conductivity λ.However, if the thermal conductivity λ is maintained constant and the heat transfer coefficient α is varied to achieve the same four Biot numbers, the outcomes will be identical to those in table 2.Moreover, the three instances have been tested with alternative angle, radius and non-convective boundary temperature values, and the results (not included here) exhibited no significant changes in terms of comparison to FEM simulation and reference to the Biot number.

Beams with internal generation that have two zero flux non-convective boundaries.
Regarding the parameter k(r), it can be inferred from table 2 and the additional testing conducted (not included here) that the model values typically approach those of FEM values when Biot number values are less than or equal to 1.However, this is not always the case -as for ∆φa with Bi = 0.5 and 1.Furthermore, for Biot numbers Bi ≥ 1, it is possible to obtain model values for k(r) that are very close to FEM values, yet the temperature values may still exhibit significant differences, as demonstrated for ∆φb with Bi = 10.

Discussion
Our quasi-one-dimensional model has been found to yield accurate temperature results for a range of Biot numbers for the case of beams with internal heat generation that have zero flux non-convective boundaries.For the case of beams with no heat generation that have non-convective boundaries kept at two distinct temperatures accurate results are achieved for Biot number value Bi ≤ 1.The Biot number is essentially the ratio of thermal conductive and convective resistances Due to 2D geometry, the resistances in equation 12 are expressed per unit of the third coordinate, the coordinate along which the heat transfer is considered zero.For the Biot number values Bi ≤ 1, convective resistance becomes larger than the conductive resistance and, even in the beams without internal heat generation, heat starts accumulating before it proceeds to the convective boundaries.And this is the model our parameter k is based on, as heat is generated in the slab (figure 2) and then leaves through convection.This is the reason why we observe a strong agreement between FEM and model temperature values for Bi ≤ 1 even for beams the parameter k is not specifically designed for.
We have not tested the third case: a beam with internal heat generation and two non-convective boundaries kept at two distinct temperatures, but we can safely assume that this case will fall somewhere between the two cases we have tested depending on which heat flux is more dominant, the one created by the internal heat generation or the one created by the temperature differences at the non-convective boundaries.

Conclusions
We have demonstrated, through comparison with FEM simulations, that the generalized quasi-onedimensional model applied to beams with uniformly increasing cross-sections accurately predicts the beam average temperature and angle-averaged temperature dependent on the beam radius.However, the prediction accuracy is limited by the choice of the parameter k -the proportionality parameter between the surface and angle-averaged temperature rises above the temperature of the fluid.The current formula for k is best fitted for the beams with heat generation that have zero flux nonconvective boundaries where the model accuracy is shown across a range of Biot numbers.For the beams with no heat generation that have non-convective boundaries kept at two distinct temperatures, the accuracy of the model is limited to Biot number Bi ≤ 1.To enhance the model's accuracy for such cases, we plan to identify a more suitable formula for k.

Figure 2 .
Figure 2. a) A rectangular slab of width h and thermal conductivity λ has two same convective boundaries characterized by heat transfer coefficient α and fluid temperature ϑ∞.Inside the slab the heat it generated uniformly.The heat flow is one-dimensional, proceeding only towards the convective boundaries.b) a corresponding beam of varying width h(r).

Figure 3 .
Figure 3.The three instances tested: a) The mean width ℎ ̅ is significantly bigger than the convective boundary ∆r, b) ℎ ̅ is appr.the same as ∆r, c) ℎ ̅ is significantly smaller than ∆r.

Table 1 .
ComparisonWe can see from table 1 that FEM and model values for the averaged temperatures and parameter k(r) are in excellent agreement across a range of Biot numbers with the largest discrepancy being just 1%.We have also done testing (not presented here) for different values of parameter ̇ and found that the values for the relative errors presented in table 1 are independent of the internal heat generation rate.
of FEM and model angle-averaged temperature  ̅ (r) through the mean relative error  ̅  ̅ () .Next are compared the FEM and model beam average temperature  ̅ through the relative error σ  ̅ .Lastly, we compare FEM and model values for parameter k(r) through the mean relative error  ̅ () .Compared are the three instances in figure3, labelled by their angle ∆φ.

Table 2 .
Comparison of FEM and model angle-averaged temperature  ̅ (r) through the mean relative error  ̅  ̅ () .Next are compared the FEM and model beam average temperature  ̅ through the relative error σ  ̅ .Lastly, we compare FEM and model values for parameter k(r) through the mean relative error  ̅ () .Compared are the three instances in figure3, labelled by their angle ∆φ.