Download ME 435: Thermal Energy Systems Design

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ME 435 Thermal Energy Systems
Lecture #18 (you may wish you had brought your
heat transfer text to class…)
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Heat Exchanger Performance
In past lectures we found that the power draw (Wc) and the capacity (Qe) are important performance
parameters for a compressor. If we can develop heat exchanger models that describe the heat transfer
rate, we have a set of equations that are coupled together. For example, the evaporator heat transfer rate
(the cooling capacity) is the same Qe that was developed for the compressor. However, we expect the
independent variables to be different for the evaporator. Also, since Qc = Wc + Qe, the condenser heat
transfer rate is coupled with the performance of both the compressor and the evaporator. In the
development of heat exchanger models, we need to determine what the independent variables are that
effect the heat transfer rate.
From our past knowledge of heat transfer, we expect that the heat transfer rate for a heat exchanger is a
function of the following parameters:




Th,i, Th,o - the hot fluid inlet/outlet temperatures
Tc,i, Tc,o - the cold fluid inlet/outlet temperatures
 h and m
 c-the hot and cold fluid mass flow rates
m
U - the overall heat transfer coefficient for the heat exchanger. Remember that this value can be
based on either the inside or outside surface areas of the heat exchanger. If we use the outside
surface area, an appropriate expression for U is:
R
1
1
1
1
1
A

 metal 
 
 Rmetal 
UA ho A
A
hi Ai
U ho
hi Ai

where A is the outside area Ao. Each term in the above equation is a resistance to heat transfer.
A - the total surface area of the heat exchanger (think of it as a sort of contact area)
Unlike the compressor, the heat exchanger performance models can be derived from fundamental heat
transfer and thermodynamics. For nearly all heat exchanger analysis, the following equations can be
written (the italicized h is enthalpy whereas the regular h is “hot”):
Q = U A (LMTD)
from heat transfer
 h (hh,i - hh,o)
Q= m
 h cp (Th,i - Th,o))
from thermodynamics (which often becomes m
 c (hc,o - hc,i)
Q =m
 c cp (Tc,o - Tc,i))
from thermodynamics (which often becomes m
In the first equation for Q, the LMTD is the log mean temperature difference - a sort of “averaged”
temperature difference throughout the heat exchanger. It is defined as:
LMTD 
T(outlet )  T(inlet )
T(outlet )
ln
T(inlet )
We will discuss the LMTD further in a minute.
Since we have adopted the notion that all energy transfers are positive, the definition of the inlet and
outlet locations must result in a positive LMTD. (WHY?)
Let’s consider the examples of a counterflow and a cocurrent flow heat exchanger. (Yup-get them
memory banks going…)
1)
2)
Thi
Tho
Thi
Tci
Tco
Tco
Tci
outlet
inlet
Tho
outlet
inlet
Group Exercise #1:
1) Draw the direction of the hot and cold flows on the diagrams above. Label which is cocurrent flow
and which is countercurrent flow. (you have 1 min.)
2) Draw an approximate diagram of the temperature profiles of the two fluids through the heat
exchangers on the diagrams below. Label them as Th,i, Th,o, Tc,i and Tc,o for each diagram. You have
5 min.
Temp
Temp
length
length
3) Write the equation for the LMTD for each type using Th,i, Th,o, Tc,i and Tc,o for each diagram in the
formula above. You have 5 min.
4) Why is it necessary to use the LMTD in heat exchanger analysis?
5) Can you think of a situation where the LMTD would not be necessary?
If any of the fluid streams remain in a single phase throughout the heat exchanger, and the temperature
change is relatively small, the enthalpy change in the thermodynamic equations can be changed to:
Cp(Th-Tc). If these assumptions are valid for both fluids in the heat exchanger, then the three
performance equations can be rewritten as
Q = U A (LMTD)
 h Cp,h(Th,i-Th,o)
Q= m
 c Cp,c(Tc,o-Tc,i)
Q= m
The trick to all of this is to manipulate the three equations above into giving you what you are after!
ALSO: There is also a special case where either
 h Cp,h = m
 c Cp,c ; the “capacity” of the two fluids is the same
1) m
2) BOTH fluids are changing phase.
For these two cases ONLY do you have Q = UA T )
-or-
From an analytical perspective, the heat transfer and thermodynamic equations fully describe the
performance of the heat exchanger. For a given heat exchanger geometry, (with A known), given heat
transfer fluids, and known U, the heat transfer and thermodynamic equations can be rearranged to
develop an equation for Q as a function of other variables. Figure 8-10 (to be handed out in class) shows
Q = f(LMTD,wh) for a specific chiller. If desired, the performance curve shown in Fig. 8-10 can be fit
using 3-D fitting techniques developed in previous lectures.
Performance of Evaporators and Condensers
Typical temperature profiles for evaporators and condensers are shown in the figures below. Notice that
in these figures, the heat exchangers are counterflow.
refrigerant
Temp
Temp
Chilled fluid
SCT
SET
coolant
refrigerant
length
length
Group Exercise #2:
1) Label which of the above is an evaporator and which is a condenser. You have 1 min.
2) What refrigerant temperatures should be used in the LMTD?
(You may want to read the following information before deciding 2).
In the evaporator and condenser the refrigerant is at the SET or SCT for 85-90% of the time.