Determination and optimisation of the winding hot-spot temperature is essential for managing the loading capability and thermal ageing of transformers. The liquid flow in the various winding cooling ducts directly impacts the position and temperature of the hot-spot.

For pump-driven oil directed (OD) cooling modes, the liquid flow distribution in the winding is governed by the winding cooling duct geometry and the winding inlet flow rate (determined by the pump and the hydraulic resistance of the complete cooling loop), while heat transfer to the liquid from the winding has a negligible influence on the flow distribution. Therefore, the hydraulic system and the thermal system can be decoupled.

**Experimental set-up**

The experimental set-up consists of a series of heated plates arranged in a rectangular section clear plastic tube construction to represent a portion of a disc-type winding, a radiator, a main tank, an expansion vessel, a flow meter, a heating unit to control winding inlet oil temperature, fittings and pipes. The winding model comprises 3 passes with 10 aluminium plates and 11 radial ducts per pass. Each aluminium plate has a dimension of 100×104×10 (mm) and is fitted into 3 mm grooves on the two side walls to make the radial duct depth 94 mm and width 104 mm. Two cartridge heaters are embedded in each plate, 30 mm from the two edges of the plate, making the distance between the two heaters 44 mm. Two thermocouples are fitted in the middle of each plate 10 mm from the two edges.

Particle image velocimetry was used to measure the liquid flow rates in the radial (horizontal) cooling ducts. Seeding particles were added to the liquid; these particles disperse evenly and follow the flow faithfully. Two laser pulses with a pre-set time interval between them were used to illuminate the seeding particles in the radial cooling ducts. Each time the laser fires, the synchroniser triggers the camera to take an image so a pair of images (frame A and frame B) is taken for the two consecutive laser pulses. Multiple image pairs are generated for each PIV measurement, and then PIV software analyses the image pairs statistically to infer the velocity profile in the radial cooling duct. Since flows in the radial cooling ducts are laminar, a quasi-parabolic velocity profile is obtained. The flow distributions shown were measured in pass 3 and the radial ducts are numbered from 1 at the bottom to 11 at the top.

**Dimensional analysis**

Dimensional analysis is used to investigate the effects of winding geometric dimensions, liquid types and total liquid flow rates on the liquid flow distributions. It is found that the dimensionless flow distribution, i.e. the volumetric flow fraction in each radial (horizontal) cooling duct is mainly controlled by the Reynolds number (*Re*) at the winding inlet and the ratio of radial duct height to axial duct width (*α*). The higher *Re* (i.e. higher liquid inlet velocity or higher liquid temperature) or higher *α*, the more uneven the flow distribution will be. When *Re* or *α* is high enough, stagnation and reverse flows at the bottom part of the winding pass can occur.

**Verification of Dimensional Analysis for Flow Distribution**

Experimental tests were conducted to verify the conclusion that by matching *Re* the dimensionless flow distribution will also be matched. The Reynolds number is defined as:

where *u _{m}* for average pass inlet velocity (m/s),

*W*for axial duct width (m),

_{duct}*ν*for kinematic viscosity (m

^{2}/s).

Since the kinematic viscosity of oil decreases with temperature, a lower oil flow rate can be compensated for by a higher oil temperature to maintain the same *Re*. To test this, three cases shown in Table 1 were measured on a benchmark winding model with radial duct height 4 mm and axial duct width 10 mm. The liquid used was mineral oil.

The average velocity distributions are different due to different total liquid flow rates, whereas the volumetric flow proportion distributions are almost identical, proving the conclusion from dimensional analysis.

**Flow Distribution for a Range of Re**

A parametric sweep of *Re* is needed to identify the quantitative relationship between the flow distribution and *Re*. It would be difficult to perform a large enough number of experimental tests to establish this relationship, so 2D CFD simulations were performed instead. Dimensionless flow distribution results from CFD simulations have been extracted and correlated to *Re *to form a predictive correlation equation.

Experimental tests, shown in Table 2, were conducted to cover the practical range of *Re* for high flow (OD) cooling modes and to verify the validity of the correlation equation. Comparisons of flow distributions from PIV measurements and those from the correlation equation are shown in the figure below. It can be seen that the calculated and measured results match closely and that with an increase of *Re*, the flow distribution gets increasingly uneven. What is more, for case 5 (highest flow *Re* = 1402), reverse flow in duct 1 at the bottom of pass 3 was shown by both the PIV measurement and the correlation equation derived from CFD simulations.

Comparison of flow distribution for Re in a practical range for OD cooling modes with vertical duct width being 10 mm. (a) Average velocity in each duct of pass 3. (b) Flow proportion in each duct of pass 3 where the total oil flow rate is regarded as one unit. The legend ‘equ’ refers to correlation equation derived from CFD simulations.**The Effect of α (radial duct height divided by axial duct width) on Flow Distribution**

The ratio of radial duct height to axial duct width, *α*, is identified by dimensional analysis and CFD parametric sweeps to influence the flow distribution in the radial ducts. The comparison of flow distributions for two cases with different *α* but with the other parameters kept the same (see Table 3) is presented the figure below. It can be seen that the calculated results from the correlation equation are close to those from PIV measurements and in the case of high *α *(narrow axial ducts or high radial ducts), the flow distribution is more uneven and reverse flow at the bottom of the pass is both predicted and observed.

Table 3: Two cases with different α’s and identical other parameters