Heat Transfer Coefficient Identification in Mini-Channel Flow Boiling with the Hybrid Picard–Trefftz Method
<p>(<b>a</b>) A view of the experimental stand; (<b>b</b>) Flow loop: ①: test section with the mini-channel (described in detail in <a href="#energies-11-02057-f002" class="html-fig">Figure 2</a>), ②: DC power supply, ③: preheater, ④: pressure control, ⑤: pump, ⑥: filter, ⑦: rotameter, and ⑧: cooler.</p> "> Figure 2
<p>(<b>a</b>) The cross-section of the test section with the mini-channel and the position of the infrared camera and high-speed camera and lighting. <span class="html-fig-inline" id="energies-11-02057-i001"> <img alt="Energies 11 02057 i001" src="/energies/energies-11-02057/article_deploy/html/images/energies-11-02057-i001.png"/></span>: the hub; <span class="html-fig-inline" id="energies-11-02057-i002"> <img alt="Energies 11 02057 i002" src="/energies/energies-11-02057/article_deploy/html/images/energies-11-02057-i002.png"/></span>: the heater; <span class="html-fig-inline" id="energies-11-02057-i003"> <img alt="Energies 11 02057 i003" src="/energies/energies-11-02057/article_deploy/html/images/energies-11-02057-i003.png"/></span>: the insulating foil; <span class="html-fig-inline" id="energies-11-02057-i004"> <img alt="Energies 11 02057 i004" src="/energies/energies-11-02057/article_deploy/html/images/energies-11-02057-i004.png"/></span>: the mini-channel with flowing liquid; <span class="html-fig-inline" id="energies-11-02057-i005"> <img alt="Energies 11 02057 i005" src="/energies/energies-11-02057/article_deploy/html/images/energies-11-02057-i005.png"/></span>: the glass lid covered the mini-channel; ⑨: an infrared camera; ⑩: a high-speed camera; ⑪: the LED lights. (<b>b</b>) The view of the mini-channel.</p> "> Figure 3
<p>The characteristic dimensions of the mini-channel.</p> "> Figure 4
<p>The block diagram of the experimental stand control system.</p> "> Figure 5
<p>The void fraction versus volumetric flow rate and heat flux.</p> "> Figure 6
<p>The test section and its schematic, illustrating the underlying model assumptions, (pictorial view, not to scale).</p> "> Figure 7
<p>(<b>a</b>) The thermogram example image; (<b>b</b>) the foil surface temperature field from the IR camera FLIR SC 7600 (FLIR Systems, Wilsonville, OR, USA) approximated using a third degree polynomial; (<b>c</b>) the flow structures; (<b>d</b>) the void fraction for <span class="html-italic">q<sub>w</sub></span> = 99.8 kW m<sup>−2</sup> and <span class="html-italic">q<sub>w</sub></span> = 153.8 kW m<sup>−2</sup>.</p> "> Figure 8
<p>The heat transfer coefficients for different heat fluxes as a function of the mini-channel length obtained using the hybrid Picard–Trefftz method.</p> ">
Abstract
:1. Introduction
2. Experiment
2.1. Experimental Stand
- NI cDAQ-9178: main module,
- NI 9211: temperature measurement (Czaki TP-201 type K thermocouples, Czaki Thermo-Product, Pruszków, Poland),
- NI 9239: voltage measurement (Kobold pressure gauges, 0–2.5 bar measurement range),
- NI 9203: current measurement (Kobold pressure drop gauge, 0–2.5 bar),
- NI 9263: adjustment of voltage to control the pumps,
- NI 9403: digital input/output to control mini-channel lighting and to trigger the thermal imaging camera.
2.2. Experimental Results
3. Mathematical Model and Methods
- A steady-state and laminar (Re < 2000) fluid flow in the mini-channel, with a constant volumetric flow rate;
- For 0 ≤ x ≤ LI, the liquid temperature in contact with the heater is equal to the saturation temperature, i.e., Tf = Tsat where the Tsat was determined by analogy to Reference [21];
- For the considered flow structures, i.e., the bubbly and bubbly-slug flow, the heat flux is transferred from the heater to the liquid phase in the proportion relative to the void fraction
- One non-zero component of the liquid velocity u(y) is parallel to the flow direction and satisfies the following condition
- The liquid temperature at the inlet of the mini-channel, Tin, is known and for x = LI, it satisfies the condition
3.1. Hybrid Picard–Trefftz Method
- In the first step, for k = 1:
- In subsequent steps, for k > 1:
3.2. One-Dimensional Approach
4. Results and Discussion
- The uncertainty of thermal conductivity: ∆λH = 0.1 Wm−1 K−1 (specified by the manufacturer);
- The accuracy of the foil temperature approximation: and = 10−4 m [25], and the uncertainty of temperature measurement is equal to = 0.55 K (specified by the manufacturer);
- The accuracy of the derivative of the approximate foil temperature with respect to y: ;
- The accuracy of the reference fluid temperature determination:
5. Conclusions
Author Contributions
Funding
Conflicts of Interest
Nomenclature
a | approximation coefficient |
B | boundary operator |
cp | specific heat, J kg−1 K−1 |
g | function |
k | iteration number |
L | length, m |
M | number of Trefftz functions |
MRD | maximum relative differences |
N | differential operator |
T | temperature, K |
p | pressure, Pa |
qw | heat flux, W m−2 |
qV | volumetric heat flux, W m−3 |
Re | Reynolds number |
u | velocity, m s−1 |
w | Trefftz function |
x | coordinate, m |
y | coordinate, m |
L2-norm | |
Greek symbols | |
α | heat transfer coefficient, W/(m2 K) |
Δ | Laplacian in Cartesian coordinates |
Δ−1 | inverse Laplacian operator |
δ | thickness; depth, m |
ε | mean relative error |
φ | void fraction |
λ | thermal conductivity, W/(m K) |
μ | dynamic viscosity, Pa s |
ρ | density, kg m−3 |
Ω | domain, m2 |
∂Ω | domain boundary, m |
Subscripts | |
approx | approximation |
ave | average |
data | measurement data |
F | foil |
f | fluid |
H | heater |
I, II, III | domain number |
loss | heat loss |
M | mini-channel |
sat | saturation |
sol | particular solution |
1D | one-dimensional approach |
2D | two-dimensional approach |
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qw (kW m−2) | 99.8 | 103.0 | 125.2 | 153.3 | 153.8 |
ε1D (%) | 1.83 | 2.36 | 1.40 | 1.26 | 1.24 |
ε2D (%) | 3.03 | 4.54 | 2.11 | 1.92 | 1.84 |
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Grabowski, M.; Hożejowska, S.; Pawińska, A.; Poniewski, M.E.; Wernik, J. Heat Transfer Coefficient Identification in Mini-Channel Flow Boiling with the Hybrid Picard–Trefftz Method. Energies 2018, 11, 2057. https://doi.org/10.3390/en11082057
Grabowski M, Hożejowska S, Pawińska A, Poniewski ME, Wernik J. Heat Transfer Coefficient Identification in Mini-Channel Flow Boiling with the Hybrid Picard–Trefftz Method. Energies. 2018; 11(8):2057. https://doi.org/10.3390/en11082057
Chicago/Turabian StyleGrabowski, Mirosław, Sylwia Hożejowska, Anna Pawińska, Mieczysław E. Poniewski, and Jacek Wernik. 2018. "Heat Transfer Coefficient Identification in Mini-Channel Flow Boiling with the Hybrid Picard–Trefftz Method" Energies 11, no. 8: 2057. https://doi.org/10.3390/en11082057