Numerical Solution of Axisymmetric Inverse Heat Conduction Problem by the Trefftz Method
"> Figure 1
<p>The schematic diagram of main loops realized in the experimental stand: 1—a test section, 2—a gear pump, 3—a compensating tank, 4—a tube-type heat exchanger, 5—a filter, 6—a mass flow meter, 7—a deaerator, 8—a data acquisition station, 9—a pc computer, 10—an ammeter, 11—a voltmeter, 12—an inverter welder, and 13—a shunt.</p> "> Figure 2
<p>The schematic diagram of the test section: 1—an annular minigap, 2—a cartridge heater, 3—a glass pipe, 4—a copper pipe, and 5—a test section header.</p> "> Figure 3
<p>The schemes of the test section: (<b>a</b>) a general view and (<b>b</b>) a cross-section with marked layers used in the mathematical model.</p> "> Figure 4
<p>Copper pipe temperature obtained with the use of 18 K-type thermocouples.</p> "> Figure 5
<p>2D fields of copper pipe temperatures and fluid temperatures obtained by the Trefftz method for: (<b>a</b>) <math display="inline"><semantics> <msup> <mi>q</mi> <mo>″</mo> </msup> </semantics></math> = 9.24 kW/m<sup>2</sup> and (<b>b</b>) <math display="inline"><semantics> <msup> <mi>q</mi> <mo>″</mo> </msup> </semantics></math> = 18.33 kW/m<sup>2</sup> (figures are not drawn to scale).</p> "> Figure 6
<p>Heat transfer coefficient vs. the distance from the minigap inlet, computed with: (<b>a</b>) formula (10) and (<b>b</b>) formula (18).</p> ">
Abstract
:1. Introduction
2. Experimental Data
3. Mathematical Model
4. Numerical Method
5. Results and Discussion
6. Conclusions
- The heat transfer coefficient values decreased along the entire heater length of the minigap in the analyzed saturated boiling region, which resulted from the increase of the vapor phase in the flow;
- The heat transfer coefficient calculated from both mathematical approaches were similar, as evidenced by the MRD. Larger MRD values were obtained for lower values of the heat flux;
- The uncertainty analysis indicated that MRE values determined from the two-dimensional approach were higher than MREs for the one-dimensional approach. MREs decreased with the increase of the heat flux, regardless of the method for determining the transfer coefficient; and
- It should be noted that using the Trefftz method is convenient because it yields solutions thatdo satisfy the governing equation exactly. This method allows solving both direct and inverse heat conduction problems.
Author Contributions
Funding
Conflicts of Interest
Nomenclature
a | thermal diffusivity, m2/s |
an, bn, cn | coefficients |
cp | specific heat capacity, J/(kg K) |
fn,0 | T-functions |
fn,i | polynomials |
g | T-function |
h | heat transfer coefficient, W/(m2 K) |
I | current, A |
J | functional |
k | thermal conductivity, W/(m K) |
L | length of a minigap, m |
N | number of T-functions |
M | number of T-functions |
MRE | mean relative error |
MRD | maximum relative differences |
MSE | mean square error |
r | radius, m |
p | pressure, hPa |
heat flux, W/m2 | |
T | temperature, K |
voltage drop, V | |
u | T-function |
v | velocity, m/s |
z | coordinate, m |
Greek Symbols | |
inverse Laplacian | |
ρ | density, kg/m3 |
Subscripts | |
approx | approximation |
ave | average |
cp | copper pipe |
f | fluid |
in | inlet |
out | outlet |
sat | saturation |
1D | one-dimensional approach |
2D | two-dimensional approach |
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Functions fn,0(r,z) | Functions gn(r,z) |
---|---|
satisfy the Laplace Equation (1) | |
recursive formulas for functions: | |
, and for n ≥ 2 | , and for n ≥ 2 |
recursive formulas for derivatives: | |
and for n ≥ 1 | , and for n ≥ 1 |
the limits when at a fixed point (r, z): | |
, | , |
Parameter | Average Value | Uncertainty |
---|---|---|
inlet fluid temperature | 293.91 K | 0.9K |
outletfluid temperature | 321.89 K | 0.9 K |
measurement temperature of the pipe surface | from 352.26 K to 371.48 K | 0.9 K |
inlet pressure | 177 kPa | ±0.05% of reading * |
outlet pressure | 167 kPa | ±0.05% of reading * |
mass flow rate | 0.0033 kg/s | ±0.10% of reading * |
voltage drop | 120 V | 0.001ΔU + 0.0001 * |
current | 1.4 A | 0.1 A * |
Boundary Conditions | Average MSE |
---|---|
condition (6) | 0.75 K |
condition (8) | 167.58 K |
condition (9) | 2.5 × 10−3 W/m2 |
(kW/m2) | 9.24 | 11.13 | 13.30 | 15.98 | 18.33 |
---|---|---|---|---|---|
Equation (10) | 15.4% | 16.5% | 14.1% | 13.5% | 14.2% |
Equation (18) | 7.1% | 7.5% | 4.9% | 6.8% | 3.5% |
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Hożejowska, S.; Piasecka, M. Numerical Solution of Axisymmetric Inverse Heat Conduction Problem by the Trefftz Method. Energies 2020, 13, 705. https://doi.org/10.3390/en13030705
Hożejowska S, Piasecka M. Numerical Solution of Axisymmetric Inverse Heat Conduction Problem by the Trefftz Method. Energies. 2020; 13(3):705. https://doi.org/10.3390/en13030705
Chicago/Turabian StyleHożejowska, Sylwia, and Magdalena Piasecka. 2020. "Numerical Solution of Axisymmetric Inverse Heat Conduction Problem by the Trefftz Method" Energies 13, no. 3: 705. https://doi.org/10.3390/en13030705