The Cooling Problem in a Very High-Power Electrical Transformer: Contribution to Heat Flow and Transfer Predictions in Innovative Technologies

Open Access Contemporary Research Analysis Journal Volume 02 Issue 11 November 2025 CrossRef DOI: 10.55677/CRAJ/03-2025-Vol02I11 e-ISSN: 3050-5909 p-ISSN: 3050-5895 Page no: 694-709 709of 694Page https://crajour.org/Available on: The Cooling Problem in a Very High-Power Electrical Transformer: Contribution to Heat Flow and Transfer Predictions in Innovative Technologies Emanuel Eduardo Pires Vaz Plant engineer, professor of applied mathematics by FEUP, IEEE, NYAS. ABSTRACT: This article provides an overview of the current state of knowledge regarding heat transfer technology in high and very high power transformers, focusing on transformer cooling, heat dissipation without insulation deterioration, and the calculation of temperatures for the hottest points of the winding. It also discusses the rotation of the heat flux field, which induces additional forces that alter the heat transfer pattern of the wall channel, causing secondary radial fluxes with high temperature gradients. Erosion of ceramic and metallic materials on the lamination surfaces is outlined. Reference is made to the cooling of the surfaces of the cooling fluid tank and cooling tubes, as well as the hottest points and temperature transients. Attention is given to heat dissipation in the cooling fluid medium. Finally, calculations of the transient response are made without considering the delay due to the transport of the insulating fluid, and the influence of this in a very high power transformer is noted. KEYWORDS: Bronze; Synthetic seawater; Passivation disruption; Buffering; Deoxygenation 1 INTRODUCTION Energy losses in a transformer manifest as heat in the core and coils and some secondary radial fluxes caused by high temperature gradients. This heat must be dissipated without allowing the windings to reach a temperature that causes excessive deterioration of the insulation. For the user, temperatures in a transformer are important for determining the amount of overload and the time of application of that overload, as well as the amount of "service life" of the transformer that will be destroyed by operation at various temperatures. The designer must be able to predict the temperature at all points in a transformer. Temperature determination and evaluation must be taken into account at all stages of the transformer's service life. The heat paths in a transformer are so complex that precise calculations of all temperatures within the transformer are not practical. Since temperature is a primary factor in determining the service life of the transformer insulation, we must pay the utmost attention to determining its value and finding ways to design it economically, within certain limits. To calculate the temperature of the hottest point in a winding, the process is as follows: • Calculation of the temperature drop in the copper through the insulation. • Calculation of the temperature drop between the coil surface and the coolant. • Calculation of the coolant temperature rise. High-power transformers are necessarily immersed in a liquid or gas, and the heat from the core and coils must flow into this fluid. We will analyze the problems of heat flow transfer in fluids involved in the design and maintenance of transformers. We present several theoretical topics and technical comments from experience on the cooling of high and very high power electrical transformers. 2. EXPERIMENTAL DETAILS In order to review the heat transfer technology for internal passages it is important to first look on the actual design of blades for transformer cooling. Figure 1 shows a typical example of such blades. The external heat transfer of the blade provides the boundary conditions for the internal cooling problem. If one considers the external heat transfer at midspan of a blade which is subjected to pure convective heat transfer without film cooling, a lot of experimental data are available. This is even true if we consider additionally roughness effects on the blade surface. If the blade is cooled additionally by applying film cooling, the situation is much more complicated. The heat flux into the surface is given by The Cooling Problem in a Very High-Power Electrical Transformer: Contribution to Heat Flow and Transfer Predictions in Innovative Technologies 709of 695Page https://crajour.org/Available on: 𝑞𝐹=ℎ𝐹(𝑇𝑎𝑤− 𝑇𝑆𝑢𝑟𝑓) (1) The heat flux qF should always be smaller than q0 , which is the heat flux without film cooling: 𝑞0 =ℎ0 (𝑇𝑟∞−𝑇𝑟𝑤) (2) Otherwise, film cooling would have a negative impact on the overall cooling of the blade. Injecting cold air into the boundary layer will basically have two different effects: On the one side, the coolant injected into the main stream will reduce the driving temperature difference for the heat transfer. This effect can be expressed, in a non dimensional form, by the adiabatic film cooling effectiveness: 𝜂𝑎𝑑= (𝑇𝑎𝑤−𝑇𝑟∞)(𝑇𝑐−𝑇𝑟∞) (3) On the other side, the cooling jets entering and disturbing the boundary layer will increase the heat transfer coefficient, so that hF is usually larger than h0 . Because of the growing importance of film cooling for blade design, the subject has been studied extensively over the past forty years and a larger part of the results is available in the open literature. Several review articles concentrate on flat plate configurations with film injection through slots, cylindrical or shaped holes. Because of the larger number of parameters influencing the film cooling, such as hole geometry, blowing and momentum flux ratio, density ratio, temperature ratio, angles of the cooling holes, curvature effects, upstream boundary layer effects, main stream turbulence effects, effect of main stream pressure gradient and the effect of surface roughness and system rotation, numerical methods and correlations have been developed to predict the adiabatic film cooling effectiveness and the increase in heat transfer coefficients. Several numerical methods can be found in literature, ref 7,13,19. See in figure 1 some theoretical topics, comments and practical technical valuable data on transformer cooling are the aim of the present study. After the external boundary conditions are known, the internal heat transfer can be discussed. The common methods for enhancing the internal heat transfer coefficients have to be chosen by the designer and are based on the cooling requirements, internal Reynolds number, and space limitations and manufacturing aspects; see ref 8,9,10,12, 15,18. As it can be seen from fig 1, the radial aligned coolant channels are connected by 180º turns. Current trends are towards increasing number of channels, so that the flow and heat transfer characteristics around bends and turns are of high interest. The Cooling Problem in a Very High-Power Electrical Transformer: Contribution to Heat Flow and Transfer Predictions in Innovative Technologies 709of 696Page https://crajour.org/Available on: Figure 2 show us flow visualization for a 180º bend. The flow characteristics in the bend region are quite complex and secondary flow and separation zones lead to complex three-dimensional flow structures, and are highly geometry dependent. 3. RESULTS AND DISCUSSION ON ANOTHER APPROACH Rotation induces additional forces on the flow field and alters the flow pattern and heat transfer distribution in the internal cooling channels. Coriolis forces introduce cross stream secondary flows, which cause different heat transfer pattern on the individual channel walls. Buoyancy forces are important at high rotational speeds and coolant temperature gradients and cause radial secondary flows. On the “high pressure side” heat transfer is generally increased above the stationary value and on the “low pressure side” heat transfer is generally decreased. See figure 3 and ref. 15. Figure 4 shows us the concept of a new combined cooling system. It consists of a heat exchanger, coolant transportation lines. Cooling air is pre-cooled and/or heat pipes are introduced to the lines, in either case, the heat exchanger is used as a heat releasing device. This system is based on the idea that coolant mass flow rate can be reduced when the coolant is pre-cooled by introducing as heat exchanger. For transformer maintenance, it is inevitable that particles floating in the atmosphere enter the engine due to the difficulty of inlet filtering. The particle motion is determined by the interaction and particle and boundary impaction. The erosion rate is strongly depended on the particle velocities, size, density and impact angles. This erosion produces oxidized aluminium particles. The Cooling Problem in a Very High-Power Electrical Transformer: Contribution to Heat Flow and Transfer Predictions in Innovative Technologies 709of 697Page https://crajour.org/Available on: If the mass ratio of erosion particles is small, the particles are known to be negligible effects on the oil flow. The particle flow field is analyzed to trace their trajectories by Lagrangian method. The governing equation of particle motion in oil flow is given by 𝑑𝑉 󰇍 󰇍 𝑃 𝑑𝜃 = 𝐶𝐷𝑅𝑒𝑃 24 𝑆𝑡𝑘(𝑉 󰇍 𝑃−𝑉 󰇍 ) (4) Where 𝐶𝐷= 24 𝑅𝑒 ← 𝑅𝑒<1 𝐶𝐷= 24 𝑅𝑒 (1+0.0916 𝑅𝑒) ← 0.1≤𝑅𝑒<5.0 𝐶𝐷= 24 𝑅𝑒 (1+0.158 𝑅𝑒2/3) ← 5.0≤𝑅𝑒 𝑆𝑡𝑘= 𝜌𝑃 𝑑𝑃 2 𝐶𝑠 18 𝜇 𝐿 To calculate the particle impact possibility on the blade, the particle impact efficiency is defined as flowtheinfloatingparticlesofnumbertotal bladetheoncollidingparticlesofnumber =  (5). In equation 5, the denominator presents the total number of particles entering the calculation domain and the numerator presents the number of particles colliding on the pressure side surface of blade. Figure 5 presents the particle impact efficiency for several flow inlet angles. As the particle size, i.e., Stokes number, increases, the particle impact efficiency has a form of step function and there is a critical value of a particle size determining whether the particle impact occurs. However, actually the particle impact efficiency cannot follow the step function as shown in fig. 5. The Cooling Problem in a Very High-Power Electrical Transformer: Contribution to Heat Flow and Transfer Predictions in Innovative Technologies 709of 698Page https://crajour.org/Available on: Figure 6 shows the erosion rates of two different materials, such as soft metal and ceramic, for the blade surface. The results show that erosion rates depend strongly on the surface material, particle impact angles, and particle sizes. In the figure, the erosion rate is given as a relative value for comparison. Erosion rates are very high along the pressure-facing blade surface for both materials. Erosion rates are very high at the leading edge due to the impact angle and the large variation in momentum of the impacting particles. The ceramic material exhibits erosion rates approximately 4 times higher at the leading edge because brittle materials generally exhibit a high erosion rate with large impact angles. Coating the blade's leading edge with ceramic material is not recommended. Published numerical calculations on convective heat transfer on the outer surfaces of blades are performed considering either a uniform wall heat flux or a uniform wall temperature: these are the boundary conditions. Significant errors can occur in the calculated loads due to the lack of consideration of internal blade cooling. Figure 7 shows a sketch of the heat transfer from the hot oil side to the inside of the blade. An energy balance for the model in Figure 8 gives: 𝑞= 𝑇2 𝑐𝑒𝑙𝑙−𝑇1𝑏 𝐷 2𝑘𝑎𝑖𝑟+1 ℎ1+𝑡𝑤𝑎𝑙𝑙 𝑘𝑤𝑎𝑙𝑙 (6) . The equations governing this study are solved by an explicit three-dimensional time-marching integration method for compressible flow using the Navier-Stokes equations. The cell-centered finite-volume method is used for discretization of the equations. For more calculation details, see reference 2 in the literature. The Cooling Problem in a Very High-Power Electrical Transformer: Contribution to Heat Flow and Transfer Predictions in Innovative Technologies 709of 699Page https://crajour.org/Available on: • The continuity equation describes conservation of mass:   0=   +  u xj j t   (7) • The momentum equation:   0)( =−+   +     jiijji j ip tuu x u (8) • The energy equation:   0)( 00 =−++   +     iji j jj j u q ueu x ep t (9) • The total internal energy, e0, is defined by: 2 0 uu ce ii vT+= (10) • The stress tensor,  ij , for compressible flow reads   ij k k i j j i ij x u x u x u   +           +  = (11) Where  ij is the Kronecker`s delta and  is the second viscosity, which is calculated as  3 2 −= . In the energy equation the heat flux vector, qj is given by Fourier’s law: x q j i T k  −= (12) The temperature distribution in the blade metal is the result of several effects: internal heat transfer by convection, external convection, and conduction through the metal itself. Consequently, although the results of external flow heat transfer alone provide important information, they do not allow for the correct evaluation of the metal temperature. An implicit Navier-Stokes solver, whose accuracy in applying heat transfer has been previously evaluated, was used for calculating the external flow. The Navier-Stokes equations are written in a general curvilinear coordinate system,  : The Cooling Problem in a Very High-Power Electrical Transformer: Contribution to Heat Flow and Transfer Predictions in Innovative Technologies 709of 700Page https://crajour.org/Available on: 𝑈𝑡+(𝐹−𝐹𝑣)𝜉+(𝐺−𝐺𝑣)𝜂=0 (13) For the vector of conservative variables and for the flux vector we have: 𝑈=𝐽−1[𝜌,𝜌𝑢,𝜌𝑣,𝑒] (14) 𝐹=𝐽−1[𝜌𝑢,𝜌𝑢𝑢,𝜉𝑥𝑝,𝜉𝑦𝑝,(𝑒+𝑝)𝑢] (15) 𝐺=𝐽−1[ρv,ρ𝑣𝑢,𝜂𝑥𝑝,𝜌𝑣𝑣,𝜂𝑦𝑝,(𝑒+𝑝)𝑣] (16) The conduction equation within the solid was discretized into a finite volume formulation on an unstructured triangular grid. The integral formulation on each triangular cell is as follows: nT c k t T..=     ∫𝜕𝑇 𝜕𝑡 Ω= ∫𝑘 𝜌 𝑐 Ω∇𝑇 𝑛 (17) The implicit discretized Euler formulation is then: ∆𝑇𝑛=∆𝑡∑𝐴𝑖1,3 𝑞1=∆𝑡∑𝐴𝑖1,3 𝜌 𝑐 𝑘 [∇(𝑇𝑛+∆𝑇𝑛)]𝑛𝑖 (18) Where ∆𝑇𝑛= 𝑇𝑛+1−𝑇𝑛 and 𝐴𝑖 is the area of cell face i. Neumann boundary conditions are easily implemented due to the finite volume integral formulation. The coupling between the fluid and the solid is obtained through an exchange of boundary conditions. The fluid flow solutions, both internal and external, are obtained with an estimated temperature distribution along the wall; these two estimators are used as a boundary condition for the solid conduction problem, resulting in a new temperature distribution along the walls. You can see a portion of such a solution to the problem in reference 11. An analytical solution to the Navier-Stokes equations in the fluid domain, Ω𝑓 is used to solve the Dirichlet boundary conditions at the interface and the Poisson conduction equations in the solid domain, Γ with Neumann boundary conditions at the solid-fluid interface: 𝑓𝑠(𝑇)=0 on Ω𝑠  𝑘𝑠𝜕𝑇 𝜕𝑛−𝑞𝑓=0 on Γ (19), (20) To solve this analytical problem applied to transformers I recommend ref. 7 where Dirichlet and Neumann problems have been treated and a particular case can be read in ref. 4. The Cooling Problem in a Very High-Power Electrical Transformer: Contribution to Heat Flow and Transfer Predictions in Innovative Technologies 709of 701Page https://crajour.org/Available on: 4. HEAT FLOW FROM OIL TO AIR WITH FORCED AIR OR FORCED - OIL COOLER. There is a significant increase in heat transfer from the tank and radiators if air is blown over the surface with cooling fans. This application is generally done empirically due to the different natures of the tank surfaces, cooling tubes, radiators, and the fans themselves, and there is no basic formula to guide the application of the fans. There are many differences in the various forced oilair cooler arrangements. Generally, two distinct arrangements are used: • Use of tubular or finned radiators, which allow a large oil flow when the pumps are off, so that the transformer can operate at 100% load without fans or pumps, and at 167% load with fans and pumps. • Use of a finned tube cooler, in which the oil and air flow is so restricted that very little heat is dissipated with pumps and fans out of operation. Typical cooling systems dissipate only about 5% of their rated capacity under these conditions, meaning that large transformers overheat just as quickly as if there were no cooling system at all. 5. TEMPERATURE TRANSIENTS AFTER LOAD CHANGES The most important factor that determines the life of a transformer's insulation is the maximum winding temperature, that is, the hottest spot. The theory presented so far has considered all temperatures as constant, that is, under steady-state conditions. However, for transient conditions, temperature calculations must take more factors into consideration, since ultimate steady-state conditions may never occur, but maximum temperatures need to be determined. Heating of conductor. The losses in the copper increases as the square of the current. For the first instant all the increment of losses goes toward heating the copper according to the following relation: 𝑇𝑒𝑚𝑝.𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒𝑠,℃= (𝑤𝑎𝑡𝑡𝑠 𝑖𝑛𝑐𝑟𝑒𝑚𝑒𝑛𝑡)(𝑡𝑖𝑚𝑒 𝑖𝑛 𝑠𝑒𝑐 ) (𝑙𝑏 𝑜𝑓 𝑐𝑜𝑝𝑝𝑒𝑟)(𝐸𝑓𝑓𝑒 𝑐𝑎𝑝) (21) Here we follow Montsinger and Clem, who determined the temperature rises and effective heat capacity of insulated copper. Effective thermal capacity=180+120Ai Ac (22) Where 𝐴𝑐= cross sectional area of conductor 𝐴𝑖= cross-sectional area of conductor insulation Substituting in eq.21 we have: 𝑇𝑒𝑚𝑝.𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒𝑠,℃= (𝑤𝑎𝑡𝑡𝑠 𝑖𝑛𝑐𝑟𝑒𝑚𝑒𝑛𝑡)(𝑡𝑖𝑚𝑒 𝑖𝑛 𝑠𝑒𝑐 ) (𝑙𝑏 𝑜𝑓 𝑐𝑜𝑝𝑝𝑒𝑟)(180+120 Ai Ac) (23) The Cooling Problem in a Very High-Power Electrical Transformer: Contribution to Heat Flow and Transfer Predictions in Innovative Technologies 709of 702Page https://crajour.org/Available on: 6. DISSIPATION OF HEAT INTO THE COOLING MEDIUM The temperature would continue to increase according to eq. 23 where it is not for heat dissipated into the cooling medium. ∆= 1 𝐾(𝑤𝑎𝑡𝑡𝑠 𝑑𝑖𝑠𝑠𝑖𝑝𝑎𝑡𝑒𝑑 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑤𝑖𝑛𝑑𝑖𝑛𝑔 𝑠𝑢𝑟𝑓𝑎𝑐𝑒)𝑛 (24) Where: n =0.8 for free convection and n = 1.0 for forced convection. K = constant of proportionality for type of cooling.  = temperature difference copper to oil ºC Equation 23 accordingly may be written: 𝑇𝑒𝑚𝑝.𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒𝑠,℃ 𝑠𝑒𝑐 =𝑟𝑎𝑡𝑒 𝑜𝑓 𝑟𝑖𝑠𝑒 𝑡𝑒𝑚𝑝= 𝑤𝑎𝑡𝑡𝑠 𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑒𝑑−𝑤𝑎𝑡𝑡𝑠 𝑑𝑖𝑠𝑠𝑖𝑝𝑎𝑡𝑒𝑑 (𝑙𝑏 𝑜𝑓 𝑐𝑜𝑝𝑝𝑒𝑟)(180+120 Ai Ac) (25) Electricity laws permits us to write: 𝑊𝑎𝑡𝑡𝑠 𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑒𝑑,𝑎𝑡 75℃= 𝐼2×(𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒,𝑎𝑡 75℃)(1+𝐾𝑒) (26) Where 𝐾𝑒=𝑟𝑎𝑡𝑖𝑜 𝑒𝑑𝑑𝑦𝑙𝑜𝑠𝑠 𝐼2×𝑅,𝑎𝑡 75℃ Watts generated (at75℃+Tc)= I2×(resistance,at 75℃)(75+Tc+234.5 75+234.5 +Ke75+234.5 75+Tc+234.5) (27) Combining equations 23 and 28 and using mathematical notation we have: dTc dt = watts generated−(KTc)1/n(area of winding) (lb of copper)(180+120Ai Ac ) (28) Where “watts generated” may be calculated by eq. 27. This equation presents a most awkward integration of n that is other than 1.0. Transformer engineers simply assume for this problem that n=1.0 and K to give an approximately correct answer. Assuming the watts generated as a constant and equal to the value corresponding to an average expected value of 𝑇𝑐, the variables can then be separated and the integration performed: 𝑇𝑐= 𝑇𝑓 𝑐−(𝑇𝑓 𝑐−𝑇𝑖 𝑐)𝑒𝑥𝑝(−𝐶) (29) The calculation of the constants gives us: 𝐶= 𝑡 𝐾×𝑎𝑟𝑒𝑎 (𝑙𝑏 𝑐𝑜𝑝𝑝𝑒𝑟)(180+120Ai Ac) (30) Where 𝑇𝑓 𝑐= watts 𝑎𝑟𝑒𝑎×𝐾 which represents the final winding rise over that would be attained at infinite time; 𝑇𝑖 𝑐= initial winding rise, and 𝐾= watts 𝑎𝑟𝑒𝑎×𝑇𝑓 𝑐 . 7. HEATING OF THE COOLING LIQUID A very good assumption for liquid-filled transformers is that the temperature of the cooling medium remains constant while the temperature of the copper increases. The liquid, however, absorbs heat from the copper and dissipates it into the air or water according to similar laws, and the temperature of the liquid will change in a manner similar to that in which the copper changed according to equation 29. The main difference is that, in the final equilibrium condition, the parts of the liquid that are in the heating and cooling zones reach, as shown, an average temperature approximately that of the oil, while a considerable mass of liquid above the coils reaches the maximum temperature of the oil. The thermal capacity of the complete transformer is much greater than the thermal capacity of the winding, and is usually expressed in watt-hours per °C, rather than watt-seconds. The final oil temperature equation is similar to equation 29: The Cooling Problem in a Very High-Power Electrical Transformer: Contribution to Heat Flow and Transfer Predictions in Innovative Technologies 709of 709Page https://crajour.org/Available on: 10. Hall, K., Johnson, B., Weigand, B. & J.Richardson. 1995. An Advanced Blade Design for V84.3 Gas Turbines, VDI Berichte, 1185: 257-275. 11. Kato, M. and Launder, B.E. 1993. The modelling of turbulent flow around stationary and vibrating square cylinders. 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