熱傳學 - HackMD

# 熱傳學 ###### tags: `academy`, `HeatTransfer`  # Ch1 Introduction to Heat Transfer ## Introduction - Energy can be transferred by **interaction** of a system with its surrounding, these **interaction** are called **work** and **heat**. - ***Thermodynamics*** deals with the end states of the thermal interaction process, but provides **no** information concerning the nature of the **interaction** and the **time rate** at which it occurs. - How much heat is transferred? - How much work is done? - Final/equilibrium staste of the system. - ![](https://i.imgur.com/ZqNWJyo.png) - ***Heat transfer*** analyzes the heat transfer rate and the modes of heat transfer due to temperature difference - How (with what modes) heat is transffered? - At what rate is heat transferred? - Temperature distruibution inside the body. - What is **heat transfer**? - Heat transfer is **thermal energy** in transit due to the temperature difference. - What is **thermal energy**? - Thermal energy is associated with the translation, rotation, vibration and electronic states of the atoms and molecules that comprise matter. It represents the cumulative effect of microscopic activities and is directly linked to the temperature of matter. | Quantity | Meaning | Symbol |Units | | -------- | -------- | -------- |--------| | Thermal Energy | Energy associated with microscopic behavior or matter | U or u |J or J/kg | | Heat Transfer| Thermal energy transport due to temperature gradiants||| | Heat | Amount of thermal energy transferred over a time interval $\Delta$t>0 | q | W| | Heat Rate| Thermal energy transfer per unit time|q|W" |Heat Flux| Thermal energy transfer per unit time and surface area.|q"|W/m^2 ## Three Modes of Heat Transfer - Conduction: Heat transfer is a sloid or a statioonary flued due the the **random motion** of its$ constinuent atoms, molecules and/or electrons. - Convection: Heat transfer due to the combined influence of **bulk and random motion** for fluid flow over a surface - Conduction and convection require the presence of temperature variations in a material medium - Radiation: **Energy** that is **emitted by matter** due to changes in the electron configurations of its atoms or molecules and is transported as electromagnetic waves(or photons). - Although radiation originates from matter, its transport does not require a material medium and occurs most efficiently in a vacuum. ## Convection - Types of convection - Forced convection: flow is caused by external means, e.g. fan, pump, wind - Natural (free) convection: flow induced by buoyancy forces due to density differences arising from temperature variations in the fluid. - Latent heat exchange associated with **phase change** -- boiling and condensation - Relation of convection to flow over a surface and development of **velocity** and **thermal boundary layers** ## Relationship to Thermodynamics :::warning Conservation of energy (first law of thermodynamics) ::: - An important tool in heat transfer analysis, often providing the **basis for determining** the **temperature** of a system. - A **closed system** is a region of fixed mass. - A **control volume(or open system) is a fixed region in space bounded by a **control surface** through which heat, work, and mass can pass. ## Special Cases(Closed system) 1. Transient Process for a Closed System of Mass > Assuming Heat Transfer to the System(Inflow) and Work Done By the System(Outflow) - Over time interval: - $E_{in} = Q$ - $E_{out} = W$ - $Q-W=\Delta E_{st}$ - For negligible changes in potential or kinetic energy - $Q-W=\Delta U_t$ (Internal thermal energy) ## Methodology of heat transfer 1. Known: State briefly what is known 2. Find: State briefly what must be found 3. Schematic: On a schematic of the system, represent the control surface by dashed line(s). Identify relevant energy transport, generation and/or storage terms by labeled arrows on the schematic. # Ch2 Introduction to Conduction ## Introduction - Conduction refers to the transport of energy in a medium(solid, liquid or gas)due to a temperature gradient. - The physical mechanism is random atomic or molecular activity - Goverend by Fourier's law ## Fourier's Law - A rate equation that allows determination of the conduction heat flux from knowledge of the temperature distribution in a medium. - Fourier's law is phenomenological, it is developed from observed phenomena - The heat flux is directional quantity - The direction of heat flow will always be normal to a surface of constant temperature, called an isothermal surface. - For an isotropic medium, the value of the thermal conductivity is independent of the coordinate direction - Implications: - Heat transfer is in the direction of decreasing temperature(basis for minus sign) - Fourier's Law serves to define the thermal conductivity of the medium - Direction of heat transfer is perpendicular to lines of constant temperature(isotherms) - Heat flux vector may be resolved into orthogonal components ## Thermal conductivity - The propotionality constant is a transport property, known us k ## Thermal Diffusivity - In heat transfer analysis, the ratio of the thermal condiuctivity to the heat capacity is an important property termed the thermal diffusivity $\alpha$ - Thermal diffusivity measures the ability of the material to conduct thermal energy relative to its ability to store thermal energy - Materails with larger $\alpha$ will respond quickly to changes in their thermal environment, while materials of small $\alpha$ will respond more sluggishly, taking longer to reach a new equilibrium condition. - It is an important parameter in Transient heat transfer analysis ## The Heat Diffusion Eqauation - A differential equation whose solution provides the temperature distribution in a stationary medium - Based on applying conservation of energy to a differential control volume through which energy transfer is exclusively by conduction - Consider: Homogeneous medium, no bulk motion, Cartesian coordinates, infinitesimally small control volume # Ch3 One-Dimensional, Steady-State Conduction - Specify appropriate form of the heat equation - Solve for the temperature distribution - Apply Fourier's law to determine the heat flux ## Contact Resistance > Due to surface roughness effects. - Conduction across the contact area and Conduction/radiation across the gaps ## One-Dimensional, Steady-State Conduction with Thermal Energy Generation **Implications of Energy Generation** - Involves a local(volumetric) source of thermal energy due to conversion from another form of energy in a conducting medium - The source may be uniformly distributed, as in the conversion from **electrical to thermal energy** - or it may be non-uniformly distributed as in the absorption of radiation passing through a semi-transparent medium - Generation affects the temperature distribution in the medium and causes the heat rate to vary with location, thereby precluding the inclusion of the medium in a thermal circuit. ## Fins performance - Fin effectiveness: The ratio of the **fin heat transfer rate** to the **heat transfer rate that would exist without the fin** # Ch4 Two-Dimensional, Steady-State Conduction ## Alternative approaches - Two dimensional conduction: - Temperature distribution is characterized by two spatial coordinates - Heat flux vector is characterized by two directional components - Heat transfer in a long, prismatic solid with two isothermal surfaces and two insulated surfaces :::info - Note the shapes of lines of constant temperature (**isotherms**) and heat flow lines(**adiabats**) - According to Fourier's law, the adiabats are everywhere **perpendicular** to isotherms ::: ## Conduction Shape Factor > Conduction shape factors and dimensionless conduction heat rates are compilations of existing solutions for geometries that are commonly encourntered in engineering practive - Two-or-three dimensional heat transfer in a medium bounded by two isothermal surfaces at $T_1$ and $T_2$ may be represented in terms of conduction shape factor S - Shape factors have been obtained analytically for numerous two and three dimensional systems, and results are summarized in Table4.1 for some common configurations. # Ch5 Transient Conduction ## Transient Conduction - A heat transfer process for which the **temperature varies with time**, as well as location within a solid. - It's initiated whenever a system experiences a **change in ooperating conditions**. - It can be induced by changes in - surface convetion conditions $(h,T_\infty)$ - surface radiation conditions $(h_r, T_\infty)$ - a surface temperature or heat flux - internal energy generation - Solution Techniques - **The Lumped Capacitance Method** - **Exact Solutions** - **The Finite-Difference Method** ## The Lumped Capacitance Method - Based on the assumption of **spatially uniform temperature distribution** throughout the transient process. Hence, $T(r,t)\approx T(t)$ - The transient problems are much more difficult than the steady state problems. The solving process is more complicated. - Temperature gradients within the solid are negligible - For a solid is in a heating or cooling process, if its volume is small and thermal conductivity is very large, thus, the slope of the temperature distribution or temperature gradient is very small within the solid. Then, the lumped capacitance method can be used to solve this problem. Even though, the solid temperature is changed with time, but at any time the solid temperature is uniform within the solid. - General Lumped Capacitance Analysis - Consider a general case, which includes convection, radiation and/or applied jheat flux at specified surfaces $(A_{S,C}, A_{S,R},A_{S,h})$ as well as internal energy generation ![](https://i.imgur.com/SIiKil7.png) ### First Law: > $\dfrac{dE_{st}}{dt}=\rho Vc\dfrac{dT}{dt}=\dot{E}_{in}-\dot{E}_{out}+\dot{E}_g$ - Assuming energy outflow due to convection and radiation and inflow due to an applied heat flux > $\rho Vc\dfrac{dT}{dt}=q''_sA_{s,h}-hA_{s,c}(T-T\infty)-h_rA_{s,r}(T-T_{sur})+\dot{E_g}$ - Special Cases (Exact Solutions) - Negligible Radiation - Lets $\theta \equiv T-T\infty$ - the time required for solid to reach some temperature T - the temperature reached byh the solid at some time t - Define Thermal time constant ## Validity of Lumped Capacitance Method To determine under what conditions it may be used with reasonable accuracy: - Assume steady-state conduction through the plane wall of area A. - One surface is maintained at a temperature $T_{s,1}$ - The other surface is exposed to a fluid of temperature $T_\infty < T_{s,1}$ - The temperature of this surface will be some intermediate value $T_{s,2}$ which $T_\infty < T_{s,2} < T_{s,1}$ The surface energy balance $\dfrac{kA}{L}(T_{s,1} - T_{s,2})=hA(T_{s,2}-T_\infty)$ $\dfrac{T_{s,1}-T_{s,2}}{T_{s,2}-T_\infty} = \dfrac{L/kA}{1/hA}=\dfrac{R_{t,cond}}{R_{t,conv}}=\dfrac{hL}{k}\equiv Bi\ \text{(The Biot Number)}$ - **The Biot number**: the first of many dimensionless parametes to be considered. - Physical Ineterpretation:$\dfrac{\text{temp diff in solid}}{\text{temp diff in fluid} = \dfrac{R_{t,cond}}{R_{t,conv}}$ - $Bi << 1$, the resistance to conduction within the solid is much less than the resistance to convection across the fluid boundary layer. Hence, the assumption of a uniform temperature distribution within the solid is reasonable if the Biot number is small. ![](https://i.imgur.com/WkPEgZH.png) $B_i \rightarrow 0, R_{t,cond} \rightarrow 0, k\rightarrow \infty , T \text{is uniform in solid}$ $if \ Bi = \dfrac{hL_c}{k}<0.1$ the error associated with using the lumped capacitance method is small $L_c=V/A_S$ definition facilitates calculation of $L_c$ for solids of complicated shape - For a plane wall of thickness 2L, $L_c=(2LA)/(2A)=$ the half-thickness $L$ - For a long cylinder, $L_c=(\pi r_0^2 L)/(2\pi r_0 L)=r_0/2$ - For a sphere, $L_c=r_0/3$ $\dfrac{\theta}{\theta_i} = \dfrac{T-T_\infty}{T_i-T_\infty}=exp\left(-\dfrac{hA_s}{\rho Vc}t\right)$ Since $L_c=V/A_s$ $\dfrac{hA_s t}{\rho V c}=\dfrac{ht}{\rho L_c c}=\dfrac{hL_c}{k}\dfrac{k}{\rho c}\dfrac{t}{L_c^2}=\dfrac{hL_c}{k}\dfrac{\alpha t}{L_c^2}=Bi \cdot Fo$ $Fo=\dfrac{\alpha t}{L_c^2} Fourier number, $a dimensionless time $\dfrac{\theta}{\theta_i}=\dfrac{T-T_\infty}{T_i-T_\infty}=exp(-Bi \cdot Fo)$ ## General Lumped Capacitance Analysis - Convection - Radiation - Thermal energy generation - Applying conservation of energy at any instant t - A nonliner, first-order, non-homogeneous, ordinary differential equation that connot be integrated to obtain an exact solution - An exact solutions may be obtained for **simplified** versions of the equation - Or can be solved numerically for a wide variety of situations involving variable properties or time-varying boundary conditions, internal ennergy generation rates, or surface heating or cooling ## Spatial Effects - For situations the Biot number is not small, temperature gradients within the medium are no longer negligible, lumped capacitance method would yield incorrect results. - Consideration must be given to spatial, as well as temporal, variations in temperature during the transient proceess. - Alternative approaches must be utilized - For a plane wall with symmetrical convection conditions and constant properties, the **heat equation** > ![](https://i.imgur.com/dWjT9jd.png) - **I.C** > $T(x,0)=T_i$ - **B.C** > $\dfrac{\partial T}{\partial x}\Big|_{x=0} = 0$ > $-k\dfrac{\partial T}{\partial x}\Big|_{x=L}=h[T(L,t)-T_\infty]$ ## Total Energy Transfer - To know the total energy that has left (or centered) the wall up to any time t in the transient process - The energy conservation requirement may be applied to time t=0 to any time t>0 - $E_{in}-E_{out}=\Delta E_{st}$ - Equating the energy transferred from the wall $Q$ to $E_{out}$, and setting $E_{in}=0$ and $\Delta E_{st}=E(t)-E(0)$, it follows that: $Q = -E_{st} = -[E(t) - E(0)] = E(0)-E(t) \\ E(t) = \rho c V T(x,t) \\ Q = -\int \rho c [T(x,t) - T_i]dV$ ## Ch6 Introduction to Convection ### Boundary Layers: Physical Features(邊界層: 物理特性) - Velocity Boundary Layer(速度邊界層) - A consequence of viscous effects associated with relative motion between a fluid and a surface > 一個因流體和表面之間的相對運動相關的黏性效應的後果 - A region of the flow characterized by shear stresses and velocity gradients > 一個以剪力和速度梯度為特徵的流動區域 - A region between the surface and the free stream whose thickness $\delta$ increases in the flow direction > 表面和自由流厚度$\delta$(Boundary Layer thickness)會隨著流的方向逐漸增大 - With increasing distance from the leading edge, the effects of viscosity penetrate farther into the free stream and thus $\delta$ increase in the flow direction > 隨與前緣的距離增加，黏度的影響會更加滲透到自由流中，因此在流動方向中$\delta$增加。 - Manifested by a surface shear stress $\tau_s$ that provides a drag force $F_D$ > 由提供拖曳力$F_D$的表面剪切應力$\tau_s$來表示 - the velocity gradient at the surface depends on the distance $x$, therefore $\tau_S$ vary in flow direction > 表面的速度梯度取決於距離x，因此$\tau_S$在流動方向上會有所不同 - Picture ![](https://i.imgur.com/rQayWwG.png) ![](https://i.imgur.com/qHFBAMs.png) - Thermal Boundary Layer(熱邊界層) - Must develop if the fluid free stream and surface temperatures differ > 如果流體自由流和表面溫度不同，則熱邊界層一定會發展(?) - A consequence of heat transfer between the surface and fluid > 熱邊界層是表面與流體進行熱傳的結果 - A region of the flow characterized by temperature gradients and heat fluxes > 一個以熱通量(heat flux)和溫度梯度為特徵的流動區域 - A region between the surface and the free stream whose thickness $\delta_t$ increases in the flow direction > 表面和自由流厚度$\delta_t$會隨著流的方向逐漸增大 - $\delta_t$ increase in the flow direction as the effects of heat transfer penetrate farther into the free stream - Manifested by a surface heat flux $q''_s$ and a convection heat transfer coefficient $h$ > 由表面熱通量$q''_s$ 和熱傳導係數$h$來表示 - at the surface, there is no fluid motion and energy transfer occurs only by conduction$q"_s = -k_f \dfrac{\partial T}{\partial y}\Bigg|_{y=0}$ > 在表面沒有流體運動，僅通過傳導發生能量轉移 - Newton's law of cooling $q"_s = h(T_s - T_\infty)$, therefore $h=\dfrac{-k_f \frac{\partial T}{\partial Y} \big|_{y=0}}{T_s - T_{\infty}}$ - Conditions in the thermal boundary layer, which strongly influence the wall temperature gradient $\frac{\partial T}{\partial y} \big|_{y=0}$, determine the rate of heat transfer across the boundary layer. > 熱邊界層的條件會嚴重影響溫度梯度，這些條件決定了跨邊界層的傳熱速率 - If $(T_S - T_\infty)$ is constant, $\frac{\partial T}{\partial y}\Big|_{y=0}$ decrease thus $q^{''}_s$ and $h$ decrease in the flow direction > 如果$(T_S - T_\infty)$是常數，則$\frac{\partial T}{\partial y}\Big|_{y=0}$減小，所以$q''_S$和$h$隨著流向方向減小 ### Significance of the Boundary Layers #### Velocity Boundary Layer - For flow over any surface, there will always exist a velocity boundary layer and hence surface friction > 對於在任何表面上的流動，始終存在速度邊界層，因此表面摩擦 - The velocity boundary layer is of extent $\delta (x)$ and is characterized by the presence of velocity gradients and shear stresses > 速度邊界層的範圍為$delta (x)$，其特徵存在於速度梯度和剪切應力。 #### Thermal Boundary Layer - A thermal boundary layer, and hence convection heat transfer, will always exist if the surface and free stream temperature differ. > 如果表面溫度和自由流溫度不同，則將始終存在熱邊界層，因此將始終存在對流傳熱。 - The thermal boundary layer is of extent, $\delta_t(x)$ and is characterized by temperature gradients and heat transfer > 熱邊界層的範圍為$\delta_t(x)$，其特徵存在於溫度低度和熱傳。 #### For the engineer - The principal manifestations of the boundary layers are, respectively, *surface friction* and *convection heat transfer*. > 邊界層的主要表現分別是表面摩擦和對流傳熱 - The key boundary layer parameters are then the *friction coefficient* $C_f$ and the *heat transfer convection coefficients* $h$, respectively. ### Local and Average Heat Transfer Coefficients - Local Heat Flux and Coefficient > $q''_S = h(T_s - T_\infty)$S > ![](https://i.imgur.com/kHAQlER.png) - Average Heat Flux and Coefficient for a Uniform Surface Temperature > $$q = \bar{h}A_S(T_S - T_\infty)$$ > $$q = \int_{A_S}''dA_S=(T_S-T_\infty)\int_{A_S}hdA_S$$ > $$\bar{h} = \dfrac{1}{A_S} \int_{A_S} hdA_S$$ - For a flat plate in parrallel flow > $$\bar{h} = \dfrac{1}{L}\int^L_0 hdx$$ ### Laminar and Turbulent Velocity Boundary Layers ![](https://i.imgur.com/RKwd3Ma.png) - In many cases, laminar and turbulent flow conditions both occur, with the laminar section preceding the turbulent section > 在許多情況下，都會出現層流(laminar)和湍流(turbulent)的條件，層流部分先於湍流部分。 - In the lamaniar boundary layer - The fluid flow is highly ordered and it is possible to identify streamlines along wich fluid particles move - The local surface shear stress $\tau_S$ decreases with increasing $x$ - In the fully turbulent boundary layer - Highly irrefular and is characterized by random, three-dimensional motion of relatively large parcels of fluid - Mixing is promoted by streamwise vortices called *streaks* within the boundary layer carries high-speed fluid toward the solid surface and transfers slower-moving fluid farther into the free stream > 邊界層內稱為「條紋」的沿流渦流促進了混合，將高速流體帶向固體表面，並將運動較慢的流體進一步轉移到自由流中 - Three different regions may be delineated within the turbulent boundary layer > 在湍流邊界層內可以劃定三個不同的區域 #### Reynolds number - $$Re_x = \dfrac{\rho u_{_\infty} x}{\mu}$$ - The onset of turbulence depends on whether the triggering mechanisms are amplified or attenuated in the direction of fluid flow, which in turn depends on - Assume transition begins at some location $x_{_c}$, critical Reynolds number for flow over a flate plate - $$Re_{_{x,c}} = \dfrac{\rho u_{_\infty}x_{_c}}{\mu} = 5 \times 10^5$$ >湍流的發生取決於觸發機制是沿流體流動方向放大還是減弱 > 假設過渡從某位置$x_{_c}$開始，這是在平板上流動的臨界雷諾數 ### Laminar and Turbulent Thermal Boundary Layers - Effect of transition on boundary layer thickness and local convection coefficient ![](https://i.imgur.com/kJCtNmU.png) - The thermal boundary layer grow in the steamwise (increasing $x$) direction, temperature gradient in the fluid at $y = 0$, the heat transfer coefficient, decrease with increasing $x$. - Turbulent mixing promotes large temperature gradient adjacent to the solid surface as well as a corresponding increase in the heat coefficients across the transition region. > 湍流混合促進了靠近固體表面的大溫度梯度，並相應地增加了過渡區域的熱係數。 - Differences in the thickness of the velocity and thermal boundary layers tend to be much smaller in turbulent flow than in laminar flow > 速度和熱邊界層厚度的差異在湍流中要比在層流中要小得多 ### The Boundary Layer Equations - Consider concurrent velocity and thermal boundary layers development for steady, two-dimensional, incompressible flow with constant fluid properties($\mu, \ c_p, \ k$) and neligible body forces. - Apply conservation of mass, Newton's $2^{\text{nd}}$ Law of Motion and conservation of energy to a differential control volume and invoke the boundary layer approximations - Two-dimensional Navier-Srokes equations for steady incompressible flow in Cartesian coordinates ### Boundary Layer Similarity - As applied to the boundary layers, the principle of similarity is based on determining similarity parameters that facilate application of results obtained for a surface experiencing one set of conditions to geometrically similar surfaces experiencing different conditions. > 當應用於邊界層時，相似性原理基於確定相似性參數，該相似性參數有利於將經歷一組條件的表面所獲得的結果應用於經歷不同條件的幾何相似表面。 - Dependent boundary layer variables of interest are $\tau_s \text{ and } q'' \text{ or } h$ > -相關的邊界層變量是$\tau_s \text{ and } q'' \text{ or } h$ - For a prescribed geometry, the corresponding independent variables are: - Geometrical: Size(L), Location(x,y) - Hydrodynamic: Velocity(V) - Fluid Properties: - Hydrodynmaic: $\rho, \mu$ - Thermal: $c_p, k$ - Key similarity parameters may be inferred by non-dimensionalizing the momentum and energy equations > 可以通過對動量和能量方程進行無量綱化來推斷關鍵相似性參數 - Recast the boundary layer equations by introducing dimensionless