Fluid and Thermal Systems
Peter Avitabile
Mechanical Engineering Department
University of Massachusetts Lowell
22.451 Dynamic Systems – Thermal/Fluid Systems
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Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Fluid and Thermal Systems
Fluid and thermal systems are generally nonlinear. However,
these systems can be linearized by evaluating them about some
normal operating point.
Fluid systems are linearized by evaluation of the system about a
normal operating point.
Thermal systems are distributed parameters and models involve
partial differential equations. Models here will have lumped
parameters so that ODE or Transfer Functions can be used.
22.451 Dynamic Systems – Thermal/Fluid Systems
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Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Liquid Level Systems
Consider a tank with cross-sectional area, A, with liquid level
height, h, and exit fluid resistance, R, density, ρ, Volume flow
in, qi, Volume flow out, qo,. The input is qi and output measured
by h.
Then
d
(ρAh ) = ρq i − ρq o
dt
qi
Rate of Change  Mass  Mass  h
 of Fluid Mass  =  flow  −  flow 

 
 

qo
Area

  in   out 
in Tank
The mass flow out can be expressed in terms of h. Assume a
linear model, the resistance of the exit orifice resistance of
the valve R is
∆p
ρq o =
R
22.451 Dynamic Systems – Thermal/Fluid Systems
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Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Liquid Level Systems
The pressure across the orifice (valve) is ∆p. The mass flow
rate out of the orifice is ρqo
p1 − p 2 (Pa + ρgh ) − Pa ρgh
⇒ ρq o =
=
=
R
R
R
where 1 − upstream, 2 − downstream
p1 − hydrostatic, p 2 − atmospheric − p a
If the tank has constant cross section, A, and assume
density, ρ, is constant, then
ρgh
dh
ρA
= ρq i −
R
dt
22.451 Dynamic Systems – Thermal/Fluid Systems
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Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Liquid Level Systems
ρgh
dh
ρA
= ρq i −
R
dt
OR
First order differential
equation with time constant
RA &
R
h + h = qi
g
g
RA
τ=
g
Note: In this problem, we are
assuming that we are operating about
some set point and are linearizing the
problem.
Note: The head versus flow rate is
not a linear function
22.451 Dynamic Systems – Thermal/Fluid Systems
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Head
g
&
Ah + h = q i
R
in differential form is
h
Operating Point
Flow Rate
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Example – Two Tank System
Pa
qi
h1
R1
A1
q1
P1
h2
R2
P2
A2
q2
P2
P3
qo
The law of conservation of mass is applied to tank 1:
d
(ρA1h1 ) = ρq i − ρq1
dt
(p a + ρgh1 ) − (p a + ρgh 2 )
p1 − p 2
= ρq i −
= ρq i −
R1
R1
Thus,
gh1 gh 2
&
A1h 1 +
−
= qi
R1 R1
22.451 Dynamic Systems – Thermal/Fluid Systems
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(System Diff. Eq. 1)
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Example – Two Tank System cont. 6-4
Conservation of mass is applied to tank 2:
d
(ρA 2 h 2 ) = ρq1 − ρq 2
dt
p1 − p 2 p 2 − p 3
=
−
R1
R2
(p a + ρgh1 ) − (p a + ρgh 2 ) (p a + ρgh 2 ) − p a
=
−
R1
R2
Thus,
gh1  1
1 
&
gh 2 = 0
A 2h 2 −
+ 
+
R1  R1 R 2 
(System Diff. Eq. 2)
22.451 Dynamic Systems – Thermal/Fluid Systems
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Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Mechanical Liquid-Level Analogy
Mechanical
Liquid-Level
Force
f
q
Flow Rate
Viscous Friction
c
c
Capacitance
Spring Constant
k
1/R
Displacement
x
h
Head
Velocity
x&
h&
Change in Head
22.451 Dynamic Systems – Thermal/Fluid Systems
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Reciprocal of Resistance
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Mechanical Liquid-Level Analogy
qi
c1
h1
h2
c2
q1
q2
c1
F
k1
x1
22.451 Dynamic Systems – Thermal/Fluid Systems
k2
c2
x2
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Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Thermal Systems: Heat Transfer
Fundamentals
Heat transfer is a branch of the thermal sciences. It can be argued that
there are only two basic and distinct modes in heat transfer: conduction and
radiation (Rohsenow and Choi, 1961). Conduction is the transfer of heat by
molecular motion. In both solids and fluids (gases and liquids), heat is
conducted by elastic collision of molecules, and energy diffusion process. The
word convection implies fluid motion. On the other hand, radiation, or more
precisely thermal radiation, belongs to the general category of electromagnetic
radiation. In thermal radiation, the energy is transmitted mainly in infrared
waves (infrared region). The mechanisms of heat transfer anywhere in the
fluid, even within solid bodies, are only conduction and radiation. The processes
of conduction and radiation occur simultaneously. In many engineering problems,
however, either conduction or radiation is dominant.
It is common practice, however, that the modes of heat transfer are
conveniently put into three groups: conduction (diffusion through a substance),
convection (fluid transport), and thermal radiation (electromagnetic radiation in
the infrared region).
22.451 Dynamic Systems – Thermal/Fluid Systems
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Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Thermal Systems: Heat Transfer
Conduction
In general, heat transfer, in an arbitrary x-direction, is governed by
the Fourier law of conduction or Fourier’s equation as
∂T
q
= −k
As
∂x
where the subscript s denotes surface, and the minus sign indicates
that the heat flow is in the direction of decreasing temperature, i.e.,
q
∂T
negative
for positive
. The equation was named after Fourier,
As
∂x
although both Biot and Fourier had suggested it. In this equation,
k =thermal conductivity of the solid or the fluid, Btu/(hr-ft-F)
T =temperature, °F
x =coordinate for an arbitrary x-direction, °F
q
=heat transfer rate per unit area, Btu/(hr-ft2)
As
22.451 Dynamic Systems – Thermal/Fluid Systems
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Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Thermal Systems: Heat Transfer
Convection
Convection usually refers to heat transfer in fluids (gas and liquids).
In this heat transfer for fluids at the solid-fluid interface is still
governed by Fourier’s equation as
 ∂Tf 
q

  = −k f 
 A s
 ∂x  s
where the subscript f indicates fluid, and s, again, stands for the
surface of the solid-fluid interface. It is often difficult to evaluate
the fluid temperature distribution, T , or the partial derivative ∂Tf at
f
∂x
the surface. It is, however, more practical to determine the heat
transfer for fluid by using Newton’s equation which was first
suggested by Newton as
q
  = h (Ts − Tf )
 A s
22.451 Dynamic Systems – Thermal/Fluid Systems
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Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Thermal Systems: Heat Transfer
Convection
In the equation
h
q
  = h (Ts − Tf )
 A s
= surface coefficient of heat transfer, Btu/(hr-ft2-F)
Ts = surface temperature, °F
Tf = fluid temperature, °F
q
  = rate of heat transfer per unit area,
 A  s evaluated at the surface, Btu/(hr-ft2)
The surface coefficient of heat transfer h, for many engineering
problems, can be found in tabulated tables in literature.
22.451 Dynamic Systems – Thermal/Fluid Systems
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Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Thermal Systems: Heat Transfer
Convection
A few words must be said about the fluid temperature. If the fluid is
infinite, Tf is the fluid temperature at a distance far away from the
surface: Tf = T∞. If the fluid is flowing in a conduit such as a pipe, T f
is the mixed-mean temperature Tf = T∞ which is defined as
R
(
Ts − T )v 2πrdr
∫
Tm = Ts − 0 R
∫0 v2πrdr
where v = v( r ) and T = T ( r ) are the fluid velocity and
temperature, respectively; both are functions of radius r; R is the
pipe radius.
22.451 Dynamic Systems – Thermal/Fluid Systems
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Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Example – Air Heating System
Assume small deviations from steady
state operation.
Assume loss to outside is negligible.
θi=
θ0=
G=
M=
c=
R=
C=
H=
θ0 + θ0
H+h
θi + θi
steady state input air, °C
steady state output air, °C
mass flow rate of air through heating chamber, kg/s
mass of air in chamber, kg
specific heat of air, kcal/kg- °C
thermal resistance, °C -s/kcal
thermal capacitance of air in heating chamber, Mc, kcal/ °C
steady state heat input, kcal/s
22.451 Dynamic Systems – Thermal/Fluid Systems
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Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Example – Air Heating System
Assume that the heat in is suddenly
changed from H to H + h and the
temperature in changes from θi to
θi + θi then the output air changes θi + θi
from θ to θ + θ.
0
0
θ0 + θ0
H+h
0
The equation to describe this behavior is
Cdθ o = [h + GC(θi − θ o )]dt
which can be rearranged as
dθ o
C
= h + GC(θi − θ o )
dt
22.451 Dynamic Systems – Thermal/Fluid Systems
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Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Example – Air Heating System
1 and substitute
GC =
R
dθ o
1
C
= h + (θi − θ o )
dt
R
dθ o 1
1
C
+ θ o = h + θi
dt
R
R
Note that
OR
dθ o
RC
+ θ o = Rh + θi
dt
First-order ODE system
-Time Constant
-Force (Input)
-Initial Condition
22.451 Dynamic Systems – Thermal/Fluid Systems
τ = RC
= Rh
= θi
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Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
qi
h
Area
Conduction
∂T
q
= −k
∂x
As
qo
RA &
R
h + h = qi
g
g
Convection
 ∂Tf 
q

  = −k f 
 A s
 ∂x  s
22.451 Dynamic Systems – Thermal/Fluid Systems
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Dr. Peter Avitabile
Modal Analysis & Controls Laboratory