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# Heat Transfer ## Modes of Heat Transfer ### 1. Conduction Heat transfer in solids, liquids, and gases due to a temperature gradient. #### Fourier's Law: $$q = -kA\frac{dT}{dx}$$ where: - $q$ is the rate of heat transfer - $k$ is the thermal conductivity of the material - $A$ is the area normal...

# Heat Transfer ## Modes of Heat Transfer ### 1. Conduction Heat transfer in solids, liquids, and gases due to a temperature gradient. #### Fourier's Law: $$q = -kA\frac{dT}{dx}$$ where: - $q$ is the rate of heat transfer - $k$ is the thermal conductivity of the material - $A$ is the area normal to the direction of heat flow - $\frac{dT}{dx}$ is the temperature gradient ### 2. Convection Heat transfer between a surface and a moving fluid. #### Newton's Law of Cooling: $$q = hA(T_s - T_{\infty})$$ where: - $q$ is the rate of heat transfer - $h$ is the convection heat transfer coefficient - $A$ is the surface area - $T_s$ is the surface temperature - $T_{\infty}$ is the fluid temperature Convection can be: - **Natural (or free) convection:** Fluid motion is due to buoyancy effects. - **Forced convection:** Fluid motion is induced by external means (e.g., a fan or pump). ### 3. Radiation Heat transfer by electromagnetic waves. #### Stefan-Boltzmann Law: $$q = \epsilon \sigma A (T_s^4 - T_{surr}^4)$$ where: - $q$ is the rate of heat transfer - $\epsilon$ is the emissivity of the surface - $\sigma$ is the Stefan-Boltzmann constant ($5.67 \times 10^{-8} W/m^2K^4$) - $A$ is the surface area - $T_s$ is the surface temperature - $T_{surr}$ is the surrounding temperature ## Thermal Resistance ### Conduction Resistance: $$R_{cond} = \frac{L}{kA}$$ ### Convection Resistance: $$R_{conv} = \frac{1}{hA}$$ ### Radiation Resistance: $$R_{rad} = \frac{1}{h_{rad}A}$$ where $h_{rad} = \epsilon \sigma (T_s + T_{surr})(T_s^2 + T_{surr}^2)$ ## Overall Heat Transfer Coefficient $$U = \frac{1}{\frac{1}{h_i} + \sum R_{cond} + \frac{1}{h_o}}$$ where: - $h_i$ is the inside heat transfer coefficient - $h_o$ is the outside heat transfer coefficient - $\sum R_{cond}$ is the sum of the conductive resistances ## Heat Exchangers ### Types: - Shell and Tube - Plate - Compact ### Log Mean Temperature Difference (LMTD): $$\Delta T_{lm} = \frac{\Delta T_1 - \Delta T_2}{\ln(\frac{\Delta T_1}{\Delta T_2})}$$ ### Heat Exchanger Effectiveness: $$\epsilon = \frac{Q_{actual}}{Q_{max}}$$ ## Fins ### Fin Efficiency: $$\eta_f = \frac{Q_{fin}}{Q_{max}}$$ ### Fin Effectiveness: $$\epsilon_f = \frac{Q_{fin}}{Q_{no fin}}$$ ## Transient Heat Conduction ### Lumped Capacitance Method: Applicable when Biot number ($Bi$) is small ($Bi < 0.1$) $$Bi = \frac{hL_c}{k}$$ where: - $h$ is the convective heat transfer coefficient - $L_c$ is the characteristic length ($L_c = \frac{V}{A_s}$) - $k$ is the thermal conductivity Temperature variation with time: $$\frac{T(t) - T_{\infty}}{T_i - T_{\infty}} = e^{-bt}$$ where: - $T(t)$ is the temperature at time $t$ - $T_i$ is the initial temperature - $T_{\infty}$ is the ambient temperature - $b = \frac{hA_s}{\rho V c_p}$ ## Two-Dimensional Steady-State Conduction ### Finite Difference Method: Discretize the domain and approximate derivatives using finite differences. ## Boiling ### Types: - Nucleate boiling - Transition boiling - Film boiling ## Condensation ### Types: - Film condensation - Dropwise condensation