Micro-channel heat sink: the model reads 0.95 to 2.09 °C hot at all four thermocouples

3Case parameters

4Experiment

Qu and Mudawar (2002) tested a copper heat sink with 21 parallel micro-channels, 231 µm wide and 713 µm deep, cooled by water and heated from below. Four thermocouples sit in the copper at x = 5, 16.588, 28.176 and 39.764 mm from the inlet, 3.175 mm above the base. I use one operating point: 100 W/cm² effective heat flux at Reynolds number 890, where the measured temperatures are 40.13 °C, 45.09 °C, 50.34 °C and 55.30 °C.

The thermocouple temperatures are not tabulated in the paper; they exist only as markers in its Fig. 7. I digitised them from the figure (marker detection on the embedded raster) and estimate the digitising uncertainty at ±0.5 °C. The paper's own figure is not reproduced here, for copyright.

Only this one operating point has a grid ladder. The paper reports five more points, which I ran on one mesh only; they are not shown.

5Set-up

One unit cell of the sink: a half copper wall either side of one channel, the channel, the polycarbonate cover above and the copper block below, 467 µm wide, 19.05 mm tall and 44.764 mm long. The side faces are symmetry (adiabatic) planes, so the cell stands for a channel in the middle of the sink. Six regions are solved together: water, copper, cover and three air pockets.

The paper's unit-cell width does not match its channel count: 467 µm against 476 µm (10 mm over 21 channels). I kept the stated power per channel, so the heat flux over the cell's footprint is 102.05 W/cm² instead of 100. The air pockets in the copper (1.6 mm wide and 3.175 mm tall at x = 10.79, 22.38 and 33.97 mm) are read from the paper's drawings; the text gives no dimensions.

The heat flux is applied on the copper part of the base only, with the pocket bottoms adiabatic, so the power per channel is conserved. The thermocouple temperature is the copper temperature at the probe position, linearly interpolated from cell centres; there is no model of the thermocouple hole or bead.

Numerics and convergence

Steady chtMultiRegionSimpleFoam, laminar, gravity off. Convection: bounded linearUpwind for momentum and energy, with an unlimited linear gradient; the same frozen settings on all four grids.

Convergence was judged on the thermocouple temperatures themselves, not on residuals or the outlet temperature: each grid ran until the largest thermocouple change was below 0.002 K per 500 iterations twice in a row. The finest grid stopped at iteration 15,000, with a last change of 0.0014 K. The outlet water temperature on the finest grid is 298.819 K.

With the linearUpwind energy scheme the water temperature falls up to 2.25 K below the inlet value near the inlet. This does not shrink with refinement (1.9-2.3 K on all four grids), so that part of the solution is not in the asymptotic range. Its effect on the copper thermocouple temperatures has not been isolated and is not inside the numerical band.

6Mesh

Every grid passes the standard checkMesh in all six regions. The extended check fails one test, small cell determinant, in the solid regions; the cells are long and flat (aspect ratio up to 142.4 in the cover), not skewed or non-orthogonal.

The coarsest grid (11,660 cells) is not a geometric refinement of the three finer grids. At TC1 it falls on the same convergence curve; at TC3 and TC4 it does not, which is why no band is given there. A fifth, similar grid (about 0.78 M cells) was not run.

7Results

At the finest of four grids (268,522 cells) the steady laminar conjugate model reads +0.95, +2.09, +1.87 and +1.42 C above the four thermocouples (digitised from Fig 7 of Qu and Mudawar 2002, +-0.5 C digitising estimate) at Re 890, 100 W/cm² planform heat flux. This is a comparison; it is not a validation.

The model is hotter everywhere, by more than the digitising estimate at every station: TC1 by about twice it, TC2 to TC4 by three to four times. The outlet water temperature is the same on all four grids to within 0.002 K and closes the energy balance to +0.13 % (21.36 W absorbed against 21.33 W input). So the water takes the right heat; the excess is in the temperature between the water and the thermocouples.

Gap and band side by side, at TC1: the model is +0.95 °C above the measurement, and the numerical uncertainty there is about 0.5 K for this scheme. At TC2, TC3 and TC4 there is no numerical band. This is a comparison, not a validation.

The paper's own unit-cell model, read from the curves in its Fig. 7, also lies above its measurements at this point. I read those curves from the raster with about 1 K error, so I use this only as a remark: the overshoot is not obviously specific to my set-up.

8Numerical accuracy

Four grids, one frozen set of numerical settings, least-squares grid-convergence estimates (Eça and Hoekstra 2014) through the shared GCI tool, with the cell size taken from the total cell count. A separate verification review set the wording below; where it refused a band, none is given.

At TC1 the four-grid least-squares estimate (Eca and Hoekstra 2014; observed order 1.5) gives a 95 % numerical uncertainty of about 0.5 K for the linearUpwind discretisation (grid term 0.24-0.58 K depending on whether cell size is taken from the fluid, whole-domain or copper cell count, plus 0.03 K iterative). The band covers grid and iteration error of this scheme only; it does not cover the choice of convection scheme or the below-inlet temperature undershoot described below.

At TC2 the four grids show no grid convergence (all four values lie within 0.155 K of each other); no numerical uncertainty is claimed.

At TC3 and TC4 no numerical uncertainty is claimed: the temperature rises steadily with refinement, but the coarsest grid is not a geometric refinement of the other three. With it, the sequence shows no convergent order, and the three similar grids alone are too few for the least-squares method.

The outlet water temperature is the same on all four grids to within 0.002 K and closes the energy balance to +0.13 % (21.36 W absorbed against 21.33 W input).

The coarsest grid (11,660 cells) is not a geometric refinement of the three finer grids. At TC1 it falls on the same convergence curve; at TC3 and TC4 it does not, which is why no band is given there. A fifth, similar grid (about 0.78 M cells) was not run.

With the linearUpwind energy scheme the water temperature falls up to 2.25 K below the inlet value near the inlet. This does not shrink with refinement (1.9-2.3 K on all four grids), so that part of the solution is not in the asymptotic range. Its effect on the copper thermocouple temperatures has not been isolated and is not inside the numerical band.

A first-order upwind run on the 96,800-cell grid gave thermocouple temperatures 0.3-1.9 C lower (largest at TC1; both runs indicative, not iteration-converged at the thermocouples). This scheme sensitivity exceeds the TC1 numerical band and is a separate, unquantified term.

LSR      Eca & Hoekstra 2014 least-squares, ng=4  (h sorted finest->coarsest: 1.0000, 1.4051, 1.9501, 2.8451)
         f: 314.233050, 314.415390, 314.790460, 315.439500
         delta_RE unweighted p =   1.5031   sigma = 0.02766
         delta_RE eq16       p =   1.5496   sigma = 0.02796
         observed order p = 1.503   (formal 2.0)   verdict: apparently monotonic
         chosen fit: p-free (unweighted)   phi0 = 313.903869   alpha = 0.319569
         sigma = 0.02766   Delta_phi = 0.4022   F_sigma = 0.069  (want < 1)
         U_phi(finest) = 0.436735  (0.139 % of f_fine; eq (20); Fs=1.25; eps=0.3196, sigma=0.02766, |f-fit|=0.009612)

LSR      Eca & Hoekstra 2014 least-squares, ng=4  (h sorted finest->coarsest: 1.0000, 1.4037, 2.0220, 2.7535)
         f: 314.233050, 314.415390, 314.790460, 315.439500
         delta_RE unweighted p =   2.0470   sigma = 0.007691
         delta_RE eq16       p =   2.0303   sigma = 0.007821
         observed order p = 2.047   (formal 2.0)   verdict: apparently monotonic
         chosen fit: p=2 (eq16)   phi0 = 314.051077   alpha = 0.182668
         sigma = 0.005989   Delta_phi = 0.4022   F_sigma = 0.015  (want < 1)
         U_phi(finest) = 0.235019  (0.075 % of f_fine; eq (20); Fs=1.25; eps=0.1827, sigma=0.005989, |f-fit|=0.0006947)
         WARNING: delta_RE NOT used: p=2.047 > 2: 4 fixed-exponent fits (App. A bullet 2). Chosen 'p=2' (eq16) on smallest sigma.

LSR      Eca & Hoekstra 2014 least-squares, ng=4  (h sorted finest->coarsest: 1.0000, 1.3947, 1.9159, 2.8855)
         f: 314.233050, 314.415390, 314.790460, 315.439500
         delta_RE unweighted p =   1.2867   sigma = 0.03814
         delta_RE eq16       p =   1.3459   sigma = 0.03861
         observed order p = 1.287   (formal 2.0)   verdict: apparently monotonic
         chosen fit: p-free (unweighted)   phi0 = 313.799944   alpha = 0.420292
         sigma = 0.03814   Delta_phi = 0.4022   F_sigma = 0.095  (want < 1)
         U_phi(finest) = 0.576319  (0.183 % of f_fine; eq (20); Fs=1.25; eps=0.4203, sigma=0.03814, |f-fit|=0.01281)

observed order p = n/a   (formal 2.0)   verdict: anomalous (p <= 0 in both delta_RE fits)

observed order p = n/a   (formal 2.0)   verdict: anomalous (p <= 0 in both delta_RE fits)

observed order p = 0.170   (formal 2.0)   verdict: apparently monotonic

9Why they differ, and what I distrust

• The measured values are digitised from a figure, not read from a table, and the digitising estimate is mine.
• Only one operating point has a grid ladder; the paper's other five points ran on one mesh here and are not shown.
• The energy scheme drives the water below its inlet temperature near the inlet, on every grid, and changing the scheme moves the thermocouples by more than the TC1 band. The band holds for this scheme only.
• Geometry read from drawings: the air pockets, the thermocouple height and the 467 µm cell width that disagrees with the channel count. Removing the pockets moved TC4 by −1.1 °C on an earlier, not thermocouple-converged run.
• A unit cell assumes all 21 channels carry the same flow and heat. The real module has inlet and outlet plenums, so flow can be maldistributed, and the heater block spreads heat in three dimensions. Both are possible causes of the excess and neither was tested.
• This compares a copper temperature, mostly set by conduction. No heat-transfer coefficient, Nusselt number or wall heat flux was compared.

10References

1. W. Qu and I. Mudawar (2002), “Experimental and numerical study of pressure drop and heat transfer in a single-phase micro-channel heat sink”, International Journal of Heat and Mass Transfer 45, 2549-2565. DOI 10.1016/S0017-9310(01)00337-4. Fig. 7 thermocouple temperatures, digitised by me.
2. L. Eça and M. Hoekstra (2014), “A procedure for the estimation of the numerical uncertainty of CFD calculations based on grid refinement studies”, Journal of Computational Physics 262, 104-130. DOI 10.1016/j.jcp.2014.01.006.
3. IAPWS-97 liquid-water properties, as fitted by my water-property script (no further citation in the local files).

Source files

• D:/CFD/data/heat-transfer/qu-mudawar-2002/fig7_tc_temperatures.csv
• D:/CFD/data/heat-transfer/qu-mudawar-2002/SOURCE.md
• D:/CFD/cht-qu-mudawar/cases/qu_q100_re890_f{0.5,0.7,1,1.4}_LU/tc_history.csv
• D:/CFD/cht-qu-mudawar/cases/qu_q100_re890_f{0.5,0.7,1,1.4}_LU/SPEC.md, results_tc.txt
• D:/CFD/cht-qu-mudawar/cases/qu_q100_re890_f1.4_LU/log.checkMesh.standard, log.checkMesh
• D:/CFD/cht-qu-mudawar/cases/qu_q100_re890_f1.4_LU/log.chtMultiRegionSimpleFoam, converge_tc.log
• D:/CFD/cht-qu-mudawar/cases/qu_q100_re890_f1.4_LU/0/, constant/
• D:/CFD/cht-qu-mudawar/queue.log
• D:/CFD/cht-qu-mudawar/GCI-baseline-q100-re890.md
• D:/CFD/cht-qu-mudawar/RESULTS-baseline-q100-re890.md
• investigation/lane-E-cht-qu-mudawar.md and lane-E2-cht-verification-review.md (evidence audit and verification review)
• pipeline/case_data/cht-qu-mudawar.py (chart data and GCI tool output)

Last updated 2026-11-14. One-page summary (prints to one A4 page), also as a PDF. Every number on this page is checked against its source file by pipeline/check_numbers.py.