5 Chapter Review Notes:
5.1 Conservation of mass
- For a closed system: the mass must remain constant - For a Control Volume mass can enter and leave the system so we must keep track of how much mass is entering and leaving the system Mass and Volume Flow Rates Mass Flow rate the amount of mass flowing through a cross section per unit time.
m Vn dAc
m m Vn dAc
Ac Ac
*Note: is used for quantities with inexact differentials, i.e. they are path functions and d is used to denote exact differentials. Average Velocity In a pipe, you never have constant velocity so you use the average velocity. 1 Vavg Vn d Ac The Mass Flow rate becomes: m Vavg Ac Ac Ac Volume Flow Rate the volume of a fluid flowing across a cross sectional area. V Vavg Ac Relation between mass flow rate and volume flow rate: V m V v Conservation of Mass Principle Conservation of mass: for a control volume, the net mass transfer to or from a control volume during a time interval, is equal to the net change in the total mass in the control volume during the time interval.
dmcv dt These equations are know as mass balance and are applicable to any control volume undergoing any kind of process. dmcv d dV Rate of change of mass in a control volume: dt dt cv - this equals 0 for a closed system min mout
min mout mcv
Mass Balance for a Steady Flow process - during steady flow, the mass in the control volume remains constant Steady Flow: min mout in these problems, we are interested in the amount of mass flowing per unit time.
The total rate of mass entering a control volume must equal the total mass leaving the flow volume. Steady Flow (single stream) m1 m2 1V1 A1 2V2 A2 - this type of flow is used in nozzles, diffusers, turbines, compressors, and pumps. Special Case: Incompressible Flow - the conservation of mass relations can be further simplified for incompressible fluids Steady incompressible flow: Vin Vout V1 V2 V1 A1 V2 A2
-
5.2 Flow Work and the energy of a flowing fluid - in a control volume, work is sometime required to push a the mass across the
control boundaries - this work is known as flow work F=pv w=fdx W flow FL PAL PV
w flow Pv
Total Energy for a flowing liquid The total energy of a a simple compressible system is as follows: V2 e u ke pe u gz 2 The total energy of a flowing fluid is as follows: Pv e Pv (u ke pe) h u pv h ke pe - Enthalpy is the flowing internal energy of a fluid - Determining flow work into and out of a system is not necessary because enthalpy takes this into account Energy Transport By Mass - you simply multiply the previous equations by m to find the energy transport due to the mass flowing into and out of a system. Amount of energy transport: E mass m mh ke pe Rate of energy Trasport: E m mh ke pe
mass
5.3 Energy Analysis of steady-flow systems
Steady flow process a process during which a volume flow through steadily this is used in turbines, nozzles, compressors, etc... - the volume, the mass, and the energy content remain constant
Mass Balance:
m
in
mout
Energy Balance: Ein Eout
dEsystem
dt this - is equal to zero for steady flow - Ein Eout This yields the following equation used in every case: 2 2 W m(h V gz) Q W m(h V gz) Q in in out out 2 2 in out = the rate of heat transfer between the control volume and its surroundings Q W =Power for constant volume, there is no boundary work!! h = enthalpy change and can be easily determined from tables - for ideal gasses h c p ,ave (T2 T1 ) ke = the kinetic energy change - this is neglected if the exit speed and the inlet speed are about the same care should be take at high velocities such as nozzles or diffusers pe = the potential energy change this is usually neglected except when used to pump a fluid up high elevations
5.4 Steady Flow Engineering Devices
Nozzles and diffusers: Nozzle a device that sincreases the velocity of a fluid at the expense of pressure Diffuser a device that increases the pressure of a fluid by slowing it down. - the rate of heat transfer is usually neglictible - work is neglictible - potential energy is neglectible - Kinetic energy is not neglected as it is the soul purpose of the device Turbines and Compressors Turbine a device that converts the flow energy for a fluid into mechanical energy through de-compression of the substance. Compressor a device that uses mechanical energy to increase the pressure in a fluid Pump a device the works like a compressor but handles liquids instead of gasses - turbines have power output and compressors have power input - Work is not neglectible - Heat transfer is neglectible - Potential energy is neglectible - Kinetic energy is usually neglictible except in the case of turbines and fans - Enthalpy change is significant!!!
Throttling Valves These devices are flow restricittng devices valves, capillary tubes, porous plugs - q=0, w=0, pe=0, ke=0 - The enthalpy remains constant during this process and is useful for calculating pressure, and temperature Isenthalpic this means that the enthalpy remains the same through this process. - for ideal gas, the temperature will remain constant Mixing Chambers Any place where two fluids are mixed - conservation of mass requires that the change in mass equal zero - q=0, w=0, pe=0, ke=0 - conservation of energy requires that the change in energy equal zero Heat Exchangers A device where two flowing fluids exchange heat without mixing. - the mass flow rate of each fluid through a heat exchanger remains constant - w=0, pe=0, ke=0 - heat is transferred with in the device but not transferred out of the device Pipe and Duct Flow - any of the attributes may be in effect in this case and the problem must be thoroughly examined to determine what is needed.
5.5 Energy Analysis of Unsteady Flow Processes
min mout msystem Uniform Flow Process the fluid flow at any inlet or outlet is uniform and steady and thus the fluid properties do not change with time or position over the cross section of an inlet or exit if they do, they are averaged and treated as constants over the entire process.
Energy balance: Qin Win m Qout Wout m m2 e2 m1e1
Q W mh mh (m2 u 2 m1u1 ) system
out in