流体力学与传热 ：1-3 Basic Equations of Fluid Flow
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Since mass carries with it associated energy due to its position, motion, or physical state, we will find that each of these types of energy will appear in the energy balance.
It is usual to take α to 1 in the calculation.
1.3.4 Overall Mechanical Energy Balance for Steady-state Flow System
The total energy balance, Eq. (4) is not often used when appreciable enthalpy changes occur or appreciable heat is added (or subtracted), since the kinetic- and potentialenergy terms are usually small and can be neglected.
a
b
At steady state the mass flow in equals the mass flow out,
m Q VaSaa VbSbb const
(1.3-6 )
The equation is also called the equation of continuity.
For an incompressible fluids
a b
If the fluid flows through a channels of circular cross section ,then the volumetric flow rate is
Q VS V d 2
4
Flow passes through two station a and b, the relationship between two velocities Va and Vb
The energy-conservation equation will then be combined with the first law of thermodynamics to obtain the final overall energy-balance equation.
Derivation of Overall Energy-Balance Equation
1.3.2 Mass Balance in a Flowing Fluid; Continuity
The principles of physics most useful in the applications of the fluid mechanics are mass-balance , or continuity ; the linearand angular-momentum-balance equation, and the mechanical-energy balance.
In addition, we can also transport energy across the boundary of the system without transferring mass
We can write the first law of thermodynamics as
E Q W
H
u 2 2
gz
Q
Ws
(4)
u2
The term 2 in equation above is the kinetic energy of a unit mass all of which
is flowing in the same velocity u.
Kinetic-energy correction factor
As a result, when appreciable heat is added or subtracted or large enthalpy changes occur, the methods for doing heat balances described are generally used.
Steady State
Streamlines
Flow along a streamline is therefore onedimensional, and a single term for velocity is all that is needed.
A stream tube is a tube of large or small cross section and of any convenient crosssection shape that is entirely bounded by streamline.
1.3 Basic Equations of Fluid Flow
In fluid dynamics fluids are in motion. They are moved from place to place by means of mechanical devices, by gravity head, or by pressure, and flow through systems of piping and/or process equipment.
The energy E in system can be classified:
Potential energy zg of a unit mass of fluid is the energy due to the position of the mass in a gravitational field g, where z is the relative height from a reference plane.
Simple Mass Balances
Consider the flow through a conduit of cross-sectional area Sa at the entrance and area Sb at the ‘exit. The average velocity and density at the entrance are Va and ρa; at the exit they are Vb and ρb
Q
Va
4
d
2 a
Vb
4
d
2 b
from which
2
Va Vb
db da
1.3-7
Where da and db are diameters of the channel at the upstream and downstream stations, respectively
1.3.3 Overall Energy Balance for Steady-state
Kinetic energy u2/2 of a unit mass of fluid is the energy present because of motion of the mass.
Internal energy U of a unit mass of a fluid is all of the other energy present, such as rotational and vibrational energy in chemical bonds.
V 2 Ek
2
u3dS 1
S
2
u3dS
S
2 m VS
VS
u3dS
S
V 3S
1.3-15
kinetic-energy can be calculated from average velocity by using αV2/2. α=2.0 for laminar flow and is about 1.05 for highly turbulent flow.
Flow System
Introduction
The principle of the conservation of energy to a control volume is much the same manner as the principle of conservation of mass.
Assuming constant density within the area S
Ek 2
u 3dS
A
5
The kinetic energy per unit mass of flowing fluid
Ek
2
u3dS 1
S
2
u3dS
S
6
m VS
VS
It is convenient to eliminate the integral by a factor operating V2/2 to give the correct value of the kinetic energy as calculated from equation 6. this factor is denoted by α and defined by
The total energy of the fluid per unit mass is then
E U u2 zg 2
1.3-9
To obtain the overall energy balance, we substitute : Eq. (1.3-9) into the entity balance Eq.(1.3-8)
When the velocity varies across the stream cross section, the kinetic energy is found in the following manner
dEk
udS u2
2
u3dS
2
4
Where Ek total flowrate of kinetic energy through the entire cross section
U u2 gz Q W
1
2
W is be divided into purely mechanical shaft work Ws and the pressure–volume work ΔPV.
py H is defined as
H U PV
3
substituting the Eqs.(2) for W and (3) for U into equation (1), and rearranging gives
One-dimensional flow
In discussing fluid flow it is helpful to visualize, in the fluid stream, fluid paths called streamlines.
A streamline is an imaginary path in a mass of flowing fluid so drawn that at every point the vector of the net velocity along the streamline u is tangent to the streamline.
1.3-8
where E is the total energy per unit mass of fluid, Q is the heat exchanged between system and environment per unit mass of fluid, and W is the work of all kinds done per unit mass of fluid, can be divided into purely mechanical shaft work and the pressure–volume work.
It is usual to take α to 1 in the calculation.
1.3.4 Overall Mechanical Energy Balance for Steady-state Flow System
The total energy balance, Eq. (4) is not often used when appreciable enthalpy changes occur or appreciable heat is added (or subtracted), since the kinetic- and potentialenergy terms are usually small and can be neglected.
a
b
At steady state the mass flow in equals the mass flow out,
m Q VaSaa VbSbb const
(1.3-6 )
The equation is also called the equation of continuity.
For an incompressible fluids
a b
If the fluid flows through a channels of circular cross section ,then the volumetric flow rate is
Q VS V d 2
4
Flow passes through two station a and b, the relationship between two velocities Va and Vb
The energy-conservation equation will then be combined with the first law of thermodynamics to obtain the final overall energy-balance equation.
Derivation of Overall Energy-Balance Equation
1.3.2 Mass Balance in a Flowing Fluid; Continuity
The principles of physics most useful in the applications of the fluid mechanics are mass-balance , or continuity ; the linearand angular-momentum-balance equation, and the mechanical-energy balance.
In addition, we can also transport energy across the boundary of the system without transferring mass
We can write the first law of thermodynamics as
E Q W
H
u 2 2
gz
Q
Ws
(4)
u2
The term 2 in equation above is the kinetic energy of a unit mass all of which
is flowing in the same velocity u.
Kinetic-energy correction factor
As a result, when appreciable heat is added or subtracted or large enthalpy changes occur, the methods for doing heat balances described are generally used.
Steady State
Streamlines
Flow along a streamline is therefore onedimensional, and a single term for velocity is all that is needed.
A stream tube is a tube of large or small cross section and of any convenient crosssection shape that is entirely bounded by streamline.
1.3 Basic Equations of Fluid Flow
In fluid dynamics fluids are in motion. They are moved from place to place by means of mechanical devices, by gravity head, or by pressure, and flow through systems of piping and/or process equipment.
The energy E in system can be classified:
Potential energy zg of a unit mass of fluid is the energy due to the position of the mass in a gravitational field g, where z is the relative height from a reference plane.
Simple Mass Balances
Consider the flow through a conduit of cross-sectional area Sa at the entrance and area Sb at the ‘exit. The average velocity and density at the entrance are Va and ρa; at the exit they are Vb and ρb
Q
Va
4
d
2 a
Vb
4
d
2 b
from which
2
Va Vb
db da
1.3-7
Where da and db are diameters of the channel at the upstream and downstream stations, respectively
1.3.3 Overall Energy Balance for Steady-state
Kinetic energy u2/2 of a unit mass of fluid is the energy present because of motion of the mass.
Internal energy U of a unit mass of a fluid is all of the other energy present, such as rotational and vibrational energy in chemical bonds.
V 2 Ek
2
u3dS 1
S
2
u3dS
S
2 m VS
VS
u3dS
S
V 3S
1.3-15
kinetic-energy can be calculated from average velocity by using αV2/2. α=2.0 for laminar flow and is about 1.05 for highly turbulent flow.
Flow System
Introduction
The principle of the conservation of energy to a control volume is much the same manner as the principle of conservation of mass.
Assuming constant density within the area S
Ek 2
u 3dS
A
5
The kinetic energy per unit mass of flowing fluid
Ek
2
u3dS 1
S
2
u3dS
S
6
m VS
VS
It is convenient to eliminate the integral by a factor operating V2/2 to give the correct value of the kinetic energy as calculated from equation 6. this factor is denoted by α and defined by
The total energy of the fluid per unit mass is then
E U u2 zg 2
1.3-9
To obtain the overall energy balance, we substitute : Eq. (1.3-9) into the entity balance Eq.(1.3-8)
When the velocity varies across the stream cross section, the kinetic energy is found in the following manner
dEk
udS u2
2
u3dS
2
4
Where Ek total flowrate of kinetic energy through the entire cross section
U u2 gz Q W
1
2
W is be divided into purely mechanical shaft work Ws and the pressure–volume work ΔPV.
py H is defined as
H U PV
3
substituting the Eqs.(2) for W and (3) for U into equation (1), and rearranging gives
One-dimensional flow
In discussing fluid flow it is helpful to visualize, in the fluid stream, fluid paths called streamlines.
A streamline is an imaginary path in a mass of flowing fluid so drawn that at every point the vector of the net velocity along the streamline u is tangent to the streamline.
1.3-8
where E is the total energy per unit mass of fluid, Q is the heat exchanged between system and environment per unit mass of fluid, and W is the work of all kinds done per unit mass of fluid, can be divided into purely mechanical shaft work and the pressure–volume work.