Lecture Note
Circuit Analysis of Dynamic Thermal Systems
-
University:
California State University, Northridge -
Course:
ME 584 | Modeling and Simulation of Dynamic Systems Academic year:
2010
-
Views:
385
Pages:
29
Author:
ete1leoje
Circuit Analysis of Dynamic Thermal
Systems Chap6
28
Model Classification
T T (t , x, y, z )
T 2T 2T 2T
f (T ,
, 2 , 2 , 2)0
t x y z
Example : Heat diffusion equation
2T 2T 2T
T
2
2
2
t
x
y
z
0
k
thermal diffusivit y
C p
T T (t )
T
f (T , ) 0
t Chap 6
29
Model Classification Tree Chp6
30
Dynamic Thermal Systems
• Dynamic response: time rate of change of
temperature during process due to large heat
capacity or small time duration
• Diffusion equation, 2nd order partial
differential equation (PDE) in space and time,
is used to model this process (beyond scope of
class)
• Take lumped-parameter model approach
under certain conditions Chp6
31
Lumped Parameter Model (LPM)
• LPM assumes that all properties of thermal resistance and
capacitance are lumped at selected points in space and
produces a set of ordinary differential equations (ODEs) in
time
• Approximating PDEs in space with ODEs in time at certain
locations in the system requires breaking system into lumps
• Choice of single-lumped or multiple-lumped capacitance
model depends on convection and conduction resistances
• Example: Model the temperature in a house
• Use a single temperature => a single ODE
• Take a representative temperature for each room => an ODE for each
room, leading to several equations
• Either model is still more manageable than if lumping were not
performed Chp6
32
Biot Number
Number of lumps required for accuracy depends on
ratio of conduction and convection resistances, i.e.,
dimensionless Biot number Nb
Rcond hLc
Nb
Rconv
k
where
Lc characteri stic length volume / surface area
Lc thickness for a plate
Lc thickness / 2 for a fin
Lc diameter / 4 for a long cylinder
Lc diameter / 6 for a sphere Chp6
33
Small Biot Number (Nb < 0.1)
• Rcond << Rconv → same
temperature in solid
• Dominant temperature
difference is between surface
and free-stream fluid
• Model with single lumped
capacitance, heat capacity of
solid and convection heat
transfer
• Ignore thermal conduction
resistance in solid Chp6
34
Large Biot Number (Nb > 0.1)
• Rcond >> Rconv → different temperatures in
solid, and between solid and free-fluid stream
• Model with several lumps of capacitance and
determine temperatures at several points
inside solid as a function of time Chp6
35
Example 5: Single-Lumped Capacitance
Model (SLCM): Quenching
• SLC model applies when heat capacity is significant (Nb is small)
• Example: Quenching is a process in which a heated object is placed into a
liquid bath. The rapid cooling that occurs can improve certain properties
such as hardness. Consider a copper sphere 1 in. in diameter with k = 212
BTU/hr-ft-0F at 5700F and immersed in a fluid such that h = 0.5 BTU/hr-ft2
(See Figure below).
– Show that its temperature can be considered uniform, and develop a model of
the sphere’s temperature as a function of the temperature T0 of the
surrounding fluid.
– Show that the RC circuit (see below) is analogous to the thermal model of the
sphere’s quenching process.
R
+
TT
T0
v0
-
C
v Chp6
36
Example 5: Single-Lumped Capacitance Model (SLCM): Quenching
• As = 4π(1/24)2=2.18x10-2 ft2, Volume = (4/3)π(1/24)3=3.03x10-4 ft3
• Lc=V/As = 0.0139, Nb = hLc/k = 3.28x10-5 << 1
• Biot criterion: treat sphere as a single-lumped system with a single
uniform T
dT
1
hA(T0 T ) (T0 T )
dt
R
slug
BTU
BTU
MC p VC p 17.3 3 (3.03 10 4 ft 3 )(2.93
)
0
.
0154
;
0
0
ft
slug F
F
(Qh ) conduction (Qh ) convection MC p
1
R
hA 0.5
1
hr 0 F
91.7
BTU
2
2
(2.18 10 ft )
BTU
hr ft 2 0 F
dv 1
dT
1
Circuit model is: C
(v0 v)
Model is : 0.0154
(T0 T )
dt R
dt 91.7
v plays the role of sphere' s temperature T
hr 0 F
BTU
RC 91.7
(0.0154 0 ) 1.4hr
BTU
F
If sphere is dropped into tank,T0 acts like a step input,
sphere's temperatur e will reach T0 in 4τ 5.6 hr Chp6
37
Multiple-Lumped Capacitance Model
(MLCM)
• Models one-dimensional heat conduction
with an equivalent thermal circuit with
lumped capacitance and resistance
– Capacitance at nodes connected by resistance
elements, analogous to mechanical system’s
discrete masses connected by massless springs
– Accuracy depends on number of lumps: large Bio
number requires division into several nodes
– Thermal properties are assumed homogeneous Chp6
38
Multiple-Lumped Capacitance Model
(MLCM)
• Models one-dimension heat transfer in a flat
plate with mass M and thickness L (<< length
and width) by dividing into n elements
Resistance elements :
1
Qh
T
Rn
L/n
Rn
kA
Capacitanc e elements:
dT
Qh Cn
dt
Cn C p M / n Chp6
39
Constant Surface Temperature Sources
• When convection coefficients are high, convection resistance
will be small, and surface temperature will equal free-stream
temperature and acts as an ideal temperature source
Resistance between Ti and Ta :
Qia
1
(Ti Ta )
Rn
Capacitanc e at Ta :
M
Ta
n
Resistance between Ta and Tb :
Qia Qab C p
Qab
1
(Ta Tb )
Rn
Capacitanc e at Tb :
M
Tb
n
Resistance between Tn 1 and T0 :
Qab Qbc C p
Qn 1, 0
1
(Tn 1 T0 )
Rn Chp6
40
Constant Surface Temperature Sources
Substitute for Q and solve for T at each node:
n
2Ta Ti Tb
Ta
C p MRn
Tb
n
2Tb Ta Tc
C p MRn
R
h0
Tn 1
n
2Tn1 Tn 2 T0
C p MRn
Use these equations to find heat transfer at any node,
e.g ., heat going into inside wall
1
Qi
(Ti Ta )
Rn Chp6
Convection Surface Temperature Sources
• When convection coefficients are not high, convection resistance will be
large and there will be convection resistance between surface
temperature and free-stream temperature, and significant temperature
difference between them
Resistance between Ti and Ti :
1
1
Qi
(Ti Ti ) where Rhi
Rhi
hi A
Capacitanc e at Ti :
Cp M
Qi Qia
Ti
2 n
Resistance between Ti and Ta :
1
Qia
(Ti Ta )
Rn
(continue on next page...)
41 Chp6
42
Convection Surface Temperature Sources
Capacitanc e at Ta :
M
Ta
n
Resistance between Tn 1 and T0 :
Qia Qab C p
Qn 1, 0
1
(Tn 1 T0 )
Rn
Capacitanc e at T0 :
Cp M
T0
2 n
Resistance between T0 and Tn 1 :
Qn 1, 0 Q0
Q0
1
1
(T0 T0 ) where Rh 0
Rh 0
h0 A Chp6
43
Convection Surface Temperature Sources
Substitute for Q and solve for T at each node:
2n
Ti
CpM
Ta
1
Ti Ta
1
Ti
Rhi Rn
Rhi Rn
n
2Ta Ti Tb
C p MRn
Tn 1
n
2Tn1 Tn2 T0
C p MRn
2n
T0
CpM
1
Tn 1 T0
1
T0
Rn
Rh 0
Rn Rh 0 Chp6
44
Example 6: Thermal Plant Boiler
Shown in the figure is a boiler used in a thermal energy
plant. Write the modeling equations for the system, and
derive a differential equation for the temperature inside the
boiler as a function of the ambient temperature and the
heat input. State the expressions for the time constant and
the steady-state temperature Chp6
45
Example 6: Thermal Plant Boiler Chp6
46
Example 7: PS: Pool Water Heater
• Shown below is a swimming pool operating in cool weather. The pool has
a heater that provides a heat input of qh watts. The water depth varies
from 1.2m to 3m, and the pool is 5m wide by 10 m long. A pump runs
continuously to keep the water thoroughly mixed.
– Calculate the Biot number, and justify the assumption that the pool can be
treated as a single-lumped capacitance model
– Assuming that the significant heat transfer is the convection at the water
surface and that no heat transfers out the sides and bottoms, write the
modeling equations for the system using a single-lumped parameter capacitor
model, and derive a differential equation for the temperature of the water as
a function of the ambient temperature and the heat input.
– Calculate the value of the time constant of the system. If a period of one time
constant would be enough time to “take the chill off,” would you recommend
leaving the heater on all the time or turning it off overnight?
– State the expression for the steady-state temperature of the water. If the
ambient temperature were 200C and the desired water temperature were
250C, what size heater would be required (in watts)? Chp6
47
Example 7: PS: Swimming Pool Heater Chp6
48
Example 7: PS: Swimming Pool Heater Chp6
49
Example 7: PS: Swimming Pool Heater Chp6
50
Example 8: Propane Tank
• A tank 75% (by volume) full of liquid propane is taken from a warehouse and
placed in the direct sunlight on a hot summer day. We are concerned about
the internal temperature (and therefore the internal pressure) of the tank as it
sits in the sun. We want to know how long it takes to heat up and what is its
final steady-state temperature.
– Model the tank as a sphere 1 foot in diameter. The walls of the tank are steel of
0.100 of an inch thick. The temperature inside the warehouse is 900F, and the
outside air temperature is 1000F. (The tank is in Texas!) The solar insolation for
this particular day is 900 watts/m2 = 285 (Btu/hr)/ft2. You should consider the sun
as a constant heat flow source of energy over the exposed surface are of the
sphere. The density of liquid propane is 20.8 lbm/ft3. The density of steel is 490
lbm/ft3. The specific heat of liquid propane is 0.58 Btu/(lbm 0F). The specific heat
of steel is 0.11 Btu/(lbm 0F). The wind velocity is such that you should use a
convection coefficient for high free convection or low forced convection.
– Model this system, and derive a differential equation for the internal temperature
of the propane as a function of the solar heat input and the ambient temperature
(with a given initial temperature). Calculate the time constant for the system, and
state how long it will take for the tank to be at its steady-state temperature. What
will the steady-state temperature be for given conditions? Chp6
51
Example 8: Propane Tank Chp6
52
Example 8: Propane Tank Chp6
53
Example 8: Propane Tank Chp6
54
Example 9: Pair-Share: Fish Tank
• A fish tank is fabricated using 4mm Plexiglas walls. The water on
the inside of the tank is maintained at a constant temperature of
260C. The ambient room temperature is 220C. We are interested in
modeling the heat transfer through the walls of the tank if there is a
sudden change in room temperature. The inside of the tank has
water that is circulated with a pump and therefore should have a
convection coefficient on the lower end of forced convection with
water. The outside of the tank has air that should have a
convection coefficient on the upper end of free convection with air.
– Calculate the Biot number for the inside and outside surfaces. Based
upon these numbers, can lumped parameter modeling be used to
model the temperature distribution in the Plexiglas? Can you assume
that the convection heat transfer is so good in the water, that the
water temperature on the inside surfaces is constant?
– Based upon the Biot number, how many lumps should be considered
in the lumped parameter analysis? Chp6
55
Example 9: Pair-Share: Fish Tank
Circuit Analysis of Dynamic Thermal Systems
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