Temperature rise and tubes for 500 kVA transformer

Only the losses and tank size are needed for this part; the core and winding data are used in the full design of the same transformer.

Basis

• A plain tank wall loses heat by radiation (≈ 6 W/m²°C) and natural convection (≈ 6.5 W/m²°C): total 12.5 W/m²°C.
• Tubes screen each other, so they add almost nothing by radiation, but they improve convection by about 35%: $6.5\times1.35 \approx 8.8$ W/m²°C.
• Top and bottom of the tank are neglected.

Temperature rise with a plain tank

Total loss $P = P_i + P_c = 1460 + 3865 = 5325$ W.

$$\begin{aligned} S_t &= 2(L_t + W_t)H_t = 2(1.05 + 0.62)\times1.6 = 5.344\ \text{m}^2\\ \theta &= \frac{P}{12.5S_t} = \frac{5325}{12.5\times5.344} = 79.7\ ^\circ\text{C} \end{aligned}$$

This is far above the permitted mean oil rise of 35 °C, so cooling tubes are needed.

Number of tubes

Let the total tube area be $A_t$:

$$\begin{aligned} P &= \theta\,(12.5S_t + 8.8A_t)\\ A_t &= \frac{1}{8.8}\left(\frac{5325}{35} - 12.5\times5.344\right) = \frac{1}{8.8}(152.14 - 66.80) = 9.698\ \text{m}^2 \end{aligned}$$

Assume tubes of 50 mm diameter and 1.5 m mean length (tank height 1.6 m):

$$\begin{aligned} \text{Area of one tube} &= \pi\times0.05\times1.5 = 0.2356\ \text{m}^2\\ n_t &= \frac{9.698}{0.2356} = 41.16 \approx 42\ \text{tubes} \end{aligned}$$

Check: $35(12.5\times5.344 + 8.8\times42\times0.2356) = 5386$ W ≥ 5325 W.

Arrangement of tubes: with 50 mm tubes at 75 mm centre-to-centre spacing, one row round the tank can hold about $1050/75 = 14$ tubes on each long side and $620/75 \approx 8$ on each short side. So 42 tubes are placed in a single row: 14 on each long side and 7 on each short side ($2\times14 + 2\times7 = 42$).

        o o o o o o o o o o o o o o    (14)
      +-------------------------------+
   o  |                               |  o
   o  |                               |  o
   o  |        TANK (top view)        |  o
(7)o  |      1.05 m  x  0.62 m        |  o (7)
   o  |                               |  o
   o  |                               |  o
   o  |                               |  o
      +-------------------------------+
        o o o o o o o o o o o o o o    (14)
   o = cooling tube, 50 mm dia, 75 mm pitch

Answer: temperature rise with plain tank ≈ 79.7 °C; 42 tubes of 50 mm diameter, 1.5 m long, keep the mean oil rise within 35 °C.