Mass & Energy Analysis of Control Volumes控制體積的質量與能量分析

What you'll be able to do本章學習成果

• Apply conservation of mass to steady- and unsteady-flow control volumes.
• Identify the energy a stream carries — internal, flow work, kinetic, potential — and fold the first two into enthalpy.
• Solve steady-flow energy balances for nozzles, turbines, compressors, throttles, mixers, and heat exchangers.
• Analyze unsteady (uniform-flow) charging and discharging processes.

Key equations重要公式

What is a control volume?什麼是控制體積？

Fluid flows continuously through a pipe, nozzle or turbine. Following one fixed lump of mass (a closed system) is awkward: it enters, passes through and leaves. Instead, fix a region in space and track everything that crosses its surface.流體持續流經管路、噴嘴或渦輪機。追蹤固定的一團質量（封閉系統）並不方便：它流入、通過、再流出。改為固定一個空間區域，追蹤所有穿越其表面的量。

• Control volume (CV): a region in space chosen for analysis. Mass, heat and work can all cross its boundary.控制體積（CV）：選定分析的空間區域。質量、熱與功皆可穿越其邊界。
• Control surface (CS): the boundary of the CV. Part of it can be real (the tube wall) and part imaginary (the inlet and exit planes).控制面（CS）：CV 的邊界。可部分為實體（管壁）、部分為假想（入口與出口截面）。
• Inlets $i$ and exits $e$: where mass crosses the CS. Here, inlet 1 and exit 2.入口 $i$ 與出口 $e$：質量穿越 CS 之處。此處為入口 1 與出口 2。
• CV vs closed system: a closed system keeps the same mass; a CV keeps the same region, so different molecules occupy it at each instant.CV 與封閉系統：封閉系統保持相同質量；CV 保持相同區域，因此每一瞬間占據其中的分子不同。
• One-dimensional flow: at each inlet and exit, properties ($p$, $T$, $v$, $\mathcal{V}$) are uniform over the area and the flow is normal to it. Assumed throughout this chapter.一維流動：每個入口與出口截面上，性質（$p$、$T$、$v$、$\mathcal{V}$）均勻，且流動垂直於截面。本章全程採此假設。

Conservation of mass質量守恆

For a closed system, mass conservation is trivial: $m$ = constant. A CV can gain or lose mass through its ports. Take a general CV with several inlets and exits, and watch it over a short time $\Delta t$.對封閉系統，質量守恆不言自明：$m$ 為常數。CV 則可經由開口增減質量。取一具多個入口與出口的一般 CV，觀察一段短時間 $\Delta t$。

Step 1 — Bookkeeping over $\Delta t$步驟 1 — $\Delta t$ 內的記帳

Mass is neither created nor destroyed, so the change of mass inside equals what came in minus what went out:質量不生不滅，因此內部質量的變化等於流入減流出：

Step 2 — Rate form步驟 2 — 速率形式

Divide by $\Delta t$ and let $\Delta t \to 0$. Each $\Delta m/\Delta t$ becomes a mass flow rate $\dot m$ (kg/s):除以 $\Delta t$ 並令 $\Delta t \to 0$。每個 $\Delta m/\Delta t$ 成為質量流率 $\dot m$（kg/s）：

For the figure, with two inlets and three exits:以圖為例，兩個入口、三個出口：

Step 3 — Mass flow rate at a port步驟 3 — 開口處的質量流率

In $\Delta t$ the slug at a port fills a cylinder of area $A$ and length $\mathcal{V}\Delta t$ (see the flow-work figure below). Its volume is在 $\Delta t$ 內，開口處的流體塊填滿截面積 $A$、長度 $\mathcal{V}\Delta t$ 的柱體（見下方流動功圖）。其體積為

Multiply by the density to get its mass:乘以密度得其質量：

Divide by $\Delta t$ and use $\rho = 1/v$:除以 $\Delta t$，並利用 $\rho = 1/v$：

Step 4 — Steady state步驟 4 — 穩態

Steady state: nothing inside the CV changes with time, so the left side of Eq. 4.2 is zero:穩態：CV 內部不隨時間變化，故式 4.2 左側為零：

For the figure:以圖為例：

One inlet, one exit: substitute $\dot m = \rho A \mathcal{V}$ on each side:單一入口、單一出口：兩側代入 $\dot m = \rho A \mathcal{V}$：

Incompressible ($\rho_1 = \rho_2$): the density cancels, so the volume flow rate is conserved too:不可壓縮（$\rho_1 = \rho_2$）：密度相消，體積流率亦守恆：

Flow work & enthalpy流動功與焓

Mass entering a CV must be pushed in against the pressure of the fluid already there. The fluid behind does the pushing, and that work is flow work. Return to the one-inlet, one-exit tube.流入 CV 的質量必須克服其中流體的壓力被推入。由後方流體推動，此功即為流動功。回到單一入口、單一出口的管路。

Step 1 — Work to push one slug in步驟 1 — 推入一個流體塊所需的功

The fluid behind the slug at inlet 1 pushes with force $F = p_1A_1$ through a distance $\mathcal{V}_1\Delta t$:入口 1 處流體塊後方的流體以力 $F = p_1A_1$ 推動，距離為 $\mathcal{V}_1\Delta t$：

Step 2 — Per unit mass步驟 2 — 換成單位質量

The product $A_1\mathcal{V}_1\Delta t$ is the slug's volume, which is also its mass times its specific volume:$A_1\mathcal{V}_1\Delta t$ 即流體塊體積，亦等於其質量乘以比容：

Substitute into Step 1, then divide by $\Delta t$:代入步驟 1，再除以 $\Delta t$：

• At an inlet, work is done on the CV: energy in.入口處，功對 CV 做：能量流入。
• At an exit, the CV pushes the fluid ahead of it: work done by the CV, $\dot m_2\,p_2v_2$, energy out.出口處，CV 推動前方流體：由 CV 做功 $\dot m_2\,p_2v_2$，能量流出。
• Flow work exists only where mass crosses the CS; none acts at the solid walls.流動功只存在於質量穿越 CS 之處；固體壁面上沒有流動功。

Net flow work done by the CV (exit minus inlet):CV 所做淨流動功（出口減入口）：

Step 3 — Bundle with internal energy: enthalpy步驟 3 — 與內能合併：焓

Every kilogram crossing a port brings its internal energy $u$ and its flow work $pv$ together. The sum is a property we already know:每公斤穿越開口的質量同時帶著內能 $u$ 與流動功 $pv$。兩者之和是已知的性質：

Control-volume energy rate balance控制體積能量速率平衡

Follow the same logic as the closed system: energy conservation for a region, now with mass also carrying energy across the boundary. Take one inlet $i$ and one exit $e$ first.沿用封閉系統的邏輯：對一區域做能量守恆，如今質量也攜帶能量穿越邊界。先取單一入口 $i$ 與單一出口 $e$。

Step 1 — Conservation over $\Delta t$步驟 1 — $\Delta t$ 內的守恆

This is the closed-system statement. Only the list of ways in and out grows.這與封閉系統的敘述相同，只是能量進出的方式變多了。

Step 2 — List every transfer步驟 2 — 列出所有傳遞方式

• Heat $Q$: as before, $Q > 0$ into the CV.熱 $Q$：同前，$Q > 0$ 為進入 CV。
• Work $W$: all work, $W > 0$ done by the CV. It splits into flow work at the ports and everything else, $W_{cv}$ (shaft, electrical, moving boundary).功 $W$：所有的功，$W > 0$ 為 CV 對外做功。可分為開口處的流動功，與其餘所有功 $W_{cv}$（軸功、電功、移動邊界功）。
• Mass: each kilogram crossing carries its stored energy (internal + kinetic + potential):質量：每公斤穿越的質量攜帶其儲存能量（內能 + 動能 + 位能）：

Step 3 — Finite form over $\Delta t$步驟 3 — $\Delta t$ 內的有限形式

Step 4 — Rate form步驟 4 — 速率形式

Divide by $\Delta t$ and let $\Delta t \to 0$:除以 $\Delta t$ 並令 $\Delta t \to 0$：

Step 5 — Separate the flow work步驟 5 — 分離流動功

From the previous section, the total work rate is the useful work plus the net flow work:由前一節，總功率為有用功加上淨流動功：

Substitute. Each $pv$ term joins the $u$ of its own stream:代入後，各 $pv$ 項併入其流體的 $u$：

Step 6 — Enthalpy and many ports步驟 6 — 焓與多個開口

Replace $u + pv$ with $h$ and sum over all inlets and exits:以 $h$ 取代 $u + pv$，並對所有入口與出口加總：

• $\dot W_{cv}$ excludes flow work; it is already inside $h$.$\dot W_{cv}$ 不含流動功，流動功已包含在 $h$ 中。

Check: close every port ($\dot m_i = \dot m_e = 0$). The sums vanish and the closed-system rate form returns:驗證：關閉所有開口（$\dot m_i = \dot m_e = 0$），加總項消失，回到封閉系統的速率形式：

Steady-flow form (SFEE)穩態流動形式（SFEE）

Step 1 — Steady state. Nothing inside the CV changes with time:步驟 1 — 穩態。CV 內部不隨時間變化：

Step 2 — One inlet, one exit. The mass balance gives步驟 2 — 單一入口、單一出口。質量平衡得

Step 3 — Substitute into Eq. 4.15 and factor out $\dot m$:步驟 3 — 代入式 4.15，並提出 $\dot m$：

Step 4 — Per unit mass. Divide by $\dot m$ to get the steady-flow energy equation:步驟 4 — 單位質量。除以 $\dot m$，得穩態流動能量方程式：

• Units: $\mathcal{V}^2/2$ and $gz$ come out in J/kg; divide by 1000 to match $h$ in kJ/kg. For example, $\mathcal{V} = 100$ m/s gives $\mathcal{V}^2/2 = 5000$ J/kg $= 5$ kJ/kg.單位：$\mathcal{V}^2/2$ 與 $gz$ 的單位為 J/kg；除以 1000 才與 kJ/kg 的 $h$ 一致。例如 $\mathcal{V} = 100$ m/s，$\mathcal{V}^2/2 = 5000$ J/kg $= 5$ kJ/kg。
• Analyzing any device is then a matter of deciding which terms are negligible.分析任何裝置，便只需判斷哪些項可忽略。

From the balance to real equipment從平衡式到實際裝置

Eq. 4.20b has five terms: heat $q$, work $w$, enthalpy change $\Delta h$, kinetic-energy change $\Delta\mathcal{V}^2/2$ and potential-energy change $g\Delta z$. A real device rarely needs all five. Before writing any numbers, go through the terms one at a time and decide whether each one matters for that device.式 4.20b 共有五項：熱 $q$、功 $w$、焓變化 $\Delta h$、動能變化 $\Delta\mathcal{V}^2/2$ 與位能變化 $g\Delta z$。實際裝置很少需要全部五項。代入數字前，應逐項檢查，判斷每一項對該裝置是否重要。

Whatever survives is the working equation for that device. For example, an insulated turbine keeps $\dot W$ and $\Delta h$; an insulated nozzle keeps $\Delta h$ and the kinetic term; a valve drops every term except $\Delta h$, which must then be zero, so $h_2 = h_1$. The next chapter, Steady-Flow Devices, applies this checklist to seven common devices one by one, with applications and typical efficiencies. The interactive SFEE device explorer is in Flow Devices & Isentropic Efficiency.剩下的項即為該裝置的工作方程式。例如：隔熱渦輪機保留 $\dot W$ 與 $\Delta h$；隔熱噴嘴保留 $\Delta h$ 與動能項；閥門除 $\Delta h$ 外各項皆可忽略，故 $\Delta h = 0$，即 $h_2 = h_1$。下一章 穩態流動裝置 將以此檢查表逐一分析七種常見裝置，並介紹其應用與典型效率。互動式 SFEE 裝置探索器請見 流動裝置與等熵效率。

Unsteady (uniform-flow) processes非穩態（均勻流動）過程

Charging or discharging a tank changes the inventory with time. Integrating the rate balances over the process gives the mass and energy accounting:

In the uniform-flow model, inlet and exit properties are taken constant, so $U_{cv}(t) - U_{cv}(0) = Q_{cv} - W_{cv} + \sum m_i h_i - \sum m_e h_e$. With all ports closed, this collapses back to the closed-system energy balance — the two views are consistent.

Try it — filling a tank動手試試——儲罐充氣

Charge a rigid, insulated tank from a high-pressure supply line. The uniform-flow energy balance gives a striking result: the gas already in the tank does compression work on the incoming gas, so the final temperature ends up above the line temperature. Fill an evacuated tank and $T_2 = k\,T_{line}$ — about 40% hotter for air.

Rigid adiabatic tank, air as ideal gas. Filled until tank pressure reaches the supply-line pressure.

Applying the control-volume energy balance控制體積能量平衡的應用

Practice problems練習題

Work each one yourself first, then reveal the solution step by step.先自行嘗試，再逐步顯示解答。