First Law for Flow Processes

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First Law of Thermodynamics for Control Volumes (Flow Processes) The First Law of Thermodynamics, also known as the conservation of energy, applied to a control volume (an open system where mass can cross the boundaries) for a steady-flow process. General Energy Balance Equation The rate form of the energy balance for a control volume is: $$ \frac{dE_{CV}}{dt} = \dot{Q}_{in} - \dot{Q}_{out} + \dot{W}_{in} - \dot{W}_{out} + \sum_{i} \dot{m}_i (h_i + \frac{V_i^2}{2} + gz_i) - \sum_{e} \dot{m}_e (h_e + \frac{V_e^2}{2} + gz_e) $$ $E_{CV}$: Total energy of the control volume $\dot{Q}$: Rate of heat transfer (positive if into CV, negative if out) $\dot{W}$: Rate of work transfer (positive if into CV, negative if out). This includes shaft work ($\dot{W}_{sh}$) and flow work. $\dot{m}$: Mass flow rate $h$: Specific enthalpy ($h = u + Pv$) $V$: Velocity $g$: Gravitational acceleration $z$: Elevation Subscripts $i$ and $e$ denote inlet and exit streams, respectively. Steady-Flow Energy Equation (SFEE) For a steady-flow process, properties within the control volume do not change with time. Thus, $\frac{dE_{CV}}{dt} = 0$. The SFEE simplifies to: $$ \dot{Q} - \dot{W}_{sh} = \sum_{e} \dot{m}_e (h_e + \frac{V_e^2}{2} + gz_e) - \sum_{i} \dot{m}_i (h_i + \frac{V_i^2}{2} + gz_i) $$ Where $\dot{W}_{sh}$ is the net shaft work done by the system (positive if work is done by the system). $\dot{Q}$ is the net heat transfer to the system. SFEE for Single-Stream (One Inlet, One Outlet) For many common devices, there is only one inlet and one outlet, and $\dot{m}_i = \dot{m}_e = \dot{m}$. $$ \dot{Q} - \dot{W}_{sh} = \dot{m} \left[ (h_e - h_i) + \frac{V_e^2 - V_i^2}{2} + g(z_e - z_i) \right] $$ Dividing by $\dot{m}$ gives the energy equation per unit mass: $$ q - w_{sh} = (h_e - h_i) + \frac{V_e^2 - V_i^2}{2} + g(z_e - z_i) $$ $q = \dot{Q}/\dot{m}$: Heat transfer per unit mass $w_{sh} = \dot{W}_{sh}/\dot{m}$: Shaft work per unit mass Common Steady-Flow Devices and Their SFEE Simplifications 1. Nozzles and Diffusers Devices that increase (nozzle) or decrease (diffuser) fluid velocity. No work, no heat transfer, negligible potential energy changes. $\dot{W}_{sh} \approx 0$, $\dot{Q} \approx 0$, $g(z_e - z_i) \approx 0$ SFEE: $0 = \dot{m} \left[ (h_e - h_i) + \frac{V_e^2 - V_i^2}{2} \right]$ Per unit mass: $h_i + \frac{V_i^2}{2} = h_e + \frac{V_e^2}{2}$ 2. Turbines and Compressors/Pumps Turbines: Produce shaft work from fluid expansion. Compressors/Pumps: Consume shaft work to increase fluid pressure. $\dot{Q} \approx 0$ (often adiabatic), $g(z_e - z_i) \approx 0$ SFEE: $-\dot{W}_{sh} = \dot{m} \left[ (h_e - h_i) + \frac{V_e^2 - V_i^2}{2} \right]$ Per unit mass: $-w_{sh} = (h_e - h_i) + \frac{V_e^2 - V_i^2}{2}$ For pumps/compressors, often $V_e \approx V_i \approx 0$. Then $-w_{sh} = h_e - h_i$. 3. Throttling Valves Devices that cause a significant pressure drop without work. Often adiabatic and negligible kinetic/potential energy changes. $\dot{W}_{sh} \approx 0$, $\dot{Q} \approx 0$, $\frac{V_e^2 - V_i^2}{2} \approx 0$, $g(z_e - z_i) \approx 0$ SFEE: $0 = \dot{m} (h_e - h_i)$ Per unit mass: $h_e = h_i$ (Isenthalpic process) 4. Mixing Chambers Where two or more streams mix to form a single stream. No work, negligible heat transfer, kinetic, or potential energy changes. $\dot{W}_{sh} \approx 0$, $\dot{Q} \approx 0$, $\Delta KE \approx 0$, $\Delta PE \approx 0$ SFEE: $\sum_{i} \dot{m}_i h_i = \sum_{e} \dot{m}_e h_e$ For two inlets and one outlet: $\dot{m}_1 h_1 + \dot{m}_2 h_2 = (\dot{m}_1 + \dot{m}_2) h_e$ 5. Heat Exchangers Transfer heat between two fluid streams without mixing. No work. The heat lost by one fluid is gained by the other (for adiabatic heat exchanger overall). $\dot{W}_{sh} \approx 0$, $\Delta KE \approx 0$, $\Delta PE \approx 0$ SFEE (for one fluid stream): $\dot{Q}_{fluid} = \dot{m}_{fluid} (h_e - h_i)$ Overall heat exchanger (adiabatic boundaries): $\sum (\dot{m}h)_{in} = \sum (\dot{m}h)_{out}$ 6. Ducts and Pipes Fluid flow through a pipe. No work. Heat transfer can occur. Kinetic and potential energy changes can be significant in some cases. $\dot{W}_{sh} \approx 0$ SFEE: $\dot{Q} = \dot{m} \left[ (h_e - h_i) + \frac{V_e^2 - V_i^2}{2} + g(z_e - z_i) \right]$ Specific Enthalpy ($h$) Defined as $h = u + Pv$, where: $u$: Specific internal energy $P$: Pressure $v$: Specific volume For ideal gases, $h = c_p T$ (if $c_p$ is constant). For incompressible liquids, $h \approx u + Pv_{avg}$.