--- title: "Natural vs artificial convection; heating curve" subject: "Electric Machine Design" chapter: 2 key: "semester_5/electric_machine_design:natural-artificial-convection" source: https://www.bctengineer.com/q/semester_5/electric_machine_design/natural-artificial-convection site: BCT Engineering --- # Natural vs artificial convection; heating curve Electric Machine Design · Chapter 2: Heating and cooling of electric machine - Asked in: 2073 Shrawan, 12 marks - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/electric_machine_design/natural-artificial-convection - Whole chapter: https://www.bctengineer.com/md/semester_5/electric_machine_design/chapter-2 ## Question Differentiate between natural and artificial convections in brief. Also derive an expression for the temperature rise-time curve for an electrical machine. ## Short answer Natural convection relies on density differences, giving low heat transfer; artificial convection uses fans/pumps for high coefficients and compact machines. The temperature rise-time curve is θ = θ_m (1 - e^{-t/τ}) + θ_i e^{-t/τ}, with θ_m = P/(Sλ) and τ = Gh/(Sλ). ## Answer Convection is the transfer of heat from a hot surface by a moving fluid (air, hydrogen, oil). It is the main way heat leaves an electrical machine. Depending on what drives the fluid, it is called natural or artificial (forced) convection. #### Natural vs artificial convection | Point | Natural convection | Artificial (forced) convection | |---|---|---| | Fluid motion caused by | Density difference: hot fluid rises, cold fluid replaces it | External agency: fan, blower or pump | | Velocity of coolant | Low | High, controlled | | Heat transfer coefficient | Low (air about 10–15 W/m²°C) | High; rises with air velocity, $\lambda \propto (1 + 0.1v)$ roughly | | Dependence on surface position | Depends on height and orientation of surface | Little dependence | | Extra equipment and losses | None | Fans/pumps; extra cost and power loss | | Reliability | Very high; no moving parts | Depends on fan/pump | | Machine size for a rating | Larger | Smaller (higher output per kg) | | Examples | Small enclosed motors, ONAN transformers | Self- and separately-ventilated motors, ONAF/OFAF transformers, hydrogen-cooled alternators | The dissipation coefficient for forced air flow is often written $\lambda_c = \lambda_{c0}(1 + 0.1\,v_a)$, where $v_a$ is the air velocity in m/s. Doubling the cooling therefore lets a much larger loss be removed for the same rise. #### Temperature rise–time curve of an electrical machine **Assumptions:** the machine is a homogeneous body with uniform temperature; losses $P$ are constant; heat dissipated is proportional to the temperature rise. **Notation:** $G$ = mass (kg), $h$ = specific heat (J/kg°C), $S$ = cooling surface (m²), $\lambda$ = specific heat dissipation (W/m²°C), $\theta$ = temperature rise at time $t$. In time $dt$, heat produced = heat stored + heat dissipated: $$ P\,dt = G h\,d\theta + S\lambda\theta\,dt $$ $$ \begin{aligned} \frac{d\theta}{dt} + \frac{S\lambda}{Gh}\theta &= \frac{P}{Gh} \end{aligned} $$ Let $\theta_m = \dfrac{P}{S\lambda}$ (final steady rise) and $\tau = \dfrac{Gh}{S\lambda}$ (heating time constant). The equation becomes $$ \tau\frac{d\theta}{dt} + \theta = \theta_m $$ Separating variables and integrating from $\theta_i$ at $t = 0$: $$ \begin{aligned} \int_{\theta_i}^{\theta}\frac{d\theta}{\theta_m - \theta} &= \int_0^t\frac{dt}{\tau} \\ \ln\frac{\theta_m - \theta_i}{\theta_m - \theta} &= \frac{t}{\tau} \\ \theta &= \theta_m\left(1 - e^{-t/\tau}\right) + \theta_i\,e^{-t/\tau} \end{aligned} $$ From cold: $\theta = \theta_m(1 - e^{-t/\tau})$. ```text rise ^ theta_m - - - - - - - - - - - - - | ___....-------~~~~ | _.-'' | .-' 0.632 theta_m at t = tau | / | / | / +---|------|------|------|------> t tau 2tau 3tau 4tau ``` **Cooling** (losses removed): $0 = Gh\,d\theta + S\lambda_c\theta\,dt$, giving $\theta = \theta_i e^{-t/\tau_c}$ with $\tau_c = Gh/(S\lambda_c)$. #### Link between convection and the curve - Forced convection raises $\lambda$. This lowers both the final rise $\theta_m = P/(S\lambda)$ and the time constant $\tau = Gh/(S\lambda)$. - For the same permitted rise, a forced-cooled machine can carry larger losses, i.e. a higher output from the same frame. - In self-ventilated machines the fan stops at standstill, so $\lambda_c < \lambda$ and the cooling time constant is longer than the heating time constant. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What drives the coolant in natural convection, compared with artificial convection? A. A fan, blower or pump B. Density difference between hot and cold fluid C. Radiation from the surface D. Conduction through the frame Answer: B. Natural convection relies on hot fluid rising; artificial convection uses an external agency. (Key point: Cause of fluid motion) 2. How does the heat transfer coefficient compare in the two types? A. Higher in natural convection B. Equal in both C. Low for forced, high for natural D. Low in natural, higher in forced and rising with air velocity Answer: D. Forced cooling raises λ roughly as (1 + 0.1v), while natural air is only about 10-15 W/m²°C. (Key point: Heat transfer coefficient) 3. Which equation results from heat produced = heat stored + heat dissipated? A. τ dθ/dt + θ = θm, with θm = P/(Sλ) and τ = Gh/(Sλ) B. dθ/dt = θ/τ C. θ = θm e^(t/τ) D. τ dθ/dt = θ Answer: A. Dividing P dt = Gh dθ + Sλθ dt by Sλ gives τ dθ/dt + θ = θm. (Key point: Heating equation) 4. Raising λ by forced convection has what effect on the heating curve? A. Raises θm and lowers τ B. Lowers θm and raises τ C. Lowers both θm and τ D. Raises both θm and τ Answer: C. Both θm = P/(Sλ) and τ = Gh/(Sλ) are inversely proportional to λ. (Key point: Effect of forced cooling) ## Related in Electric Machine Design - [Temperature rise-time curve under heating](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machine_design%3Aheating-curve-time-constant) — Electric Machine Design, chapter 2 - [Machine temperature after one hour of heating](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machine_design%3Amachine-temperature-after-hour) — Electric Machine Design, chapter 2 - [Transformer heating and cooling time constants (25°C/37.5°C)](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machine_design%3Atransformer-heating-cooling-constants) — Electric Machine Design, chapter 2, asked 3× - [Transformer heating and cooling time constants (25°C/40°C)](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machine_design%3Atransformer-cooling-constant-45c) — Electric Machine Design, chapter 2, asked 3× - [Temperature rise and tubes for 500 kVA transformer](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machine_design%3Atank-tubes-500kva-single-phase) — Electric Machine Design, chapter 3, asked 2× - [True/False: ampere conductors and temperature rise](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machine_design%3Atf-ampere-conductor-temperature) — Electric Machine Design, chapter 5 ## Related in other subjects - [Temperature rise of machine field winding](https://www.bctengineer.com/question.md?key=semester_1%2Fbasic_electrical_engineering%3Afield-winding-temperature-rise-160-185) — Basic Electrical Engineering, chapter 1 - [Natural versus artificial seasoning](https://www.bctengineer.com/question.md?key=semester_3%2Fcivil_engineering_materials%3Anatural-artificial-seasoning) — Civil Engineering Materials, chapter 7 - [Natural and forced convection compared](https://www.bctengineer.com/question.md?key=semester_5%2Fheat_transfer%3Afree-versus-forced-convection) — Heat Transfer, chapter 3 - [Natural/artificial lighting, daylight factor numerical](https://www.bctengineer.com/question.md?key=semester_6%2Fbuilding_technology%3Adaylight-numerical-062p) — Building Technology, chapter 1