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decoupled modeling of chilled-water cooling coils.

by:Mingfa Tech     2020-01-26
Abstract freezing
Water-cooled coils are important components in air
Processing system.
Typically, the cooling coil removes moisture and heat from entering the air.
Because the sensing mode and latentheat transmission mode are coupled, the saturated humidity ratio is
The temperature curve on the Psych wet graph is non-linear, and it is very difficult to solve the differential equation of heat transfer of the cooling coil on the whole coil.
However, the constant sensible heat ratio can be used by the heat and latent heat transfer modes hook remover (SHR)
, Saturated humidity ratio.
The temperature curve can be treated as linear in a small area corresponding to the coil finite element.
Then, using the validity, it is easy to derive the element equation from the decoupled explicit heat transfer and latent heat transfer differential equationNTUmethod.
In this paper, a decoupling cooling coil model is established using the finite element method and applied to the simulation of a specific cooling coil case.
Introduction modeling of cold-
The water coil plays an important role in analyzing the operation of the Air processor and fault detection and diagnosis.
Can use asteady-simulate cooling coil performance
State or dynamic model of different applications. Thesteady-
In most cases, the state model is sufficient to simulate the thermal performance of the cooling coil (Chow 1997). Basic steady-
Differential equations or control equations using state heat transfer and mass transfer in steady state calculations
State cooling coil model.
Heat and Mass Transfer in the cooling coil includes air-
Lateral sensation and latent heat transfer (
From wet air to the surface of the coil)and water-
Side heat transfer (
From the surface of the coil to the chilled water).
The governing equations are discussed in detail by Mirth and Ramadhyani (1993)andby Khan (1994).
Because the ratio of apparent and latent heat transfer mode and saturated humidity
The temperature curve is non-linear and it is very difficult to solve these differential equations on the whole coil.
In order to get the solution of these equations, several simplified analysis models are established.
The simplified model uses the heat exchanger analogy theory, which is perfectly applicable to coils with only sensible heat transfer.
Effectiveness
NTU methods in heat exchanger analogy theory are well known such as Incropera and DeWitt (2002)
, ASHRAE gives a detailed solution for coils with heat-sensitive transfer only (2000).
However, some assumptions must be applied to simplify these coupled heat transfer differential equations to conform to the standard format in the heat exchanger analogy theory.
Mirth and Ramadhyani (1993)
Several simplified analysis models are compared.
Elmahdy and Mitalas (1977)
Developed a single
A potential model using a differential value as the sole driving force for calculating total heat transfer.
This model assumes the slope of saturated specific heat, enthalpy-
The saturation temperature curve along the entire coil is constant and the Lewis number is uniform.
A single potential model recommended by ASHRAE (ASHRAE 2000;
Bourdouxhe, etc. 1998). McQuiston (1975, 1978)
Developed double
Using the potential model of temperature difference to drive the sensible heat transfer and the humidity ratio to drive the latent heat transfer.
This model assumes that the entire cooling process line is a straight line on the psychrometric chart, and that the SHRS corresponding to the entire coil are constant.
In this model, SHRS are used to decoupling induction and delay along the entire coil.
It is well known that the cooling coil can work in partial wet conditions, the inlet is dry, and the outlet of the coil is damp.
Then, when the air passes through the coil, the SHR becomes smaller as the unit changes.
Wet-saturated specific heat variable, on the other hand
Temperature of bulb due to nonlinear enthalpy-
The humidity table on the Saturationtemperature curve (ASHRAE 2001).
The saturated specific heat changes from 0. 5 to 1. 1 Btu/lb x [degrees]F in awet-
[Between 40 bulb temperature range]degrees]F and 80[degrees]F.
Due to these assumptions, there may be errors in the simplified approach.
These assumptions are not used by numerical methods. Khan (1994)
The partial performance of the cooling coil was analyzed using a numerical modelLoad condition.
The cooling coil is reduced to 1-
Size counter current air-
Instead of the actual two-water heat exchanger
Size Cross
Flow heat exchanger.
It is then divided into multiple control volumes along the airflow, one of which-
Converting dimensional differential equations into finite-
The difference equation within each control volume.
Because these finite difference equations are derived from complex control equations rather than using the existing heat exchanger analogy theory, even if the coil is reduced to one-
Size configuration.
In this paper, a numerical cooling coil model is developed using the validity.
The NTU method is then solved using the finite element method.
First, the basic differential equation for heat transfer of cooling coils is given.
Then, assuming the constant value of shrs within each element and the slope of the saturation curve, the explicit heat transfer mode of the cooling coil is decoupled from the latent heat transfer mode;
The SHR value represents the element condition, whether it is completely wet, completely dry or partially wet.
After that, both the sensible heat transfer mode and the latent heat transfer mode have the standard format of the heat exchanger analogy theory.
Finally, the element equation is obtained directly for the heat exchange and latent heat transfer of thesensible using theeffectiveness-
Big Cross
Rather than solving complex differential equations.
Both SHRS and city slopes are determined by unknown conditions for air, coilsurface, and cold water, so an iterative approach must be used.
It first assumes the slope values of the SHRS and curves, and then determines the air, coil, and freezing-
Water conditions.
Then, update the SHRS and curve slope values to the values corresponding to the calculated conditions and repeat the calculations until the values of all elements converge.
The model is applied to the cooling simulation under different conditions.
Modeling the cooling coil control equation usually, the water pipe of the cooling coil is arranged in multiple
Row configuration loop in Cross
Upstream arrangement (ASHRAE 2000).
The hydropower Road has equal supply and return through the supply and return head.
The tube can be staggered or placed in a straight line with respect to the airflow.
Coil with in-
There is parallel flow through the line arrangement, and there are three coils with stage arrangement-
Dimension water passes through.
Figure 1a shows in-
Line cooling coil with fins.
The inlet air is at right angles to the tube surface of the coil, and the tube surface is also located in the outlet box position of the oil.
The air outlet is at the relative surface of the coil where the corresponding inlet collection box is located.
Coil can be divided into multiple elements based on the number of coilrow in both dimensions.
Each line can contain one or more elements.
The easiest way is to treat each row as an element, as shown in Figure 1b.
Each element has four nodes, two on the air side and two on the water side.
The node number is circled on the water side to distinguish the node number on the air side.
Each element can be treated as a small cross
As shown in Figure 1c, the flow heat exchanger without fins.
Sensitive and potential heat transfer occurs between wet air and the surface of the undecorated pipe on the air side, and heat transfer occurs between the surface of the undecorated pipe and the chilled water.
On the other hand, in order to follow the water loop, the static cooling coil must be treated in three dimensions.
Because the air and water flow are horizontal with the cooling coil elements, as shown in Figure 1c, if the change of conduction temperature is ignored, the vertical heat transfer between air and water through conduction does not need to be described in the model.
Therefore, the element model of the staggered coil with this assumption is still the same
A 3D modelline coil. [
Figure 1 slightly]
Dry air and water vapor are treated as a mixture of ideal gases, and the heat transfer coefficient and mass transfer coefficient are treated as constants when the control equation is derived.
When in contact with the surface of the cooler, the air loses its heat.
Heat Transfer in differential surface area dA can be expressed as d [q. sub. as]= -[dot. m. sub. a][c. sub. pa]d[t. sub. a]=[U. sub. a]([t. sub. a]-[t. sub. s])dA, (1)where [dot. m. sub. a]
Dry flow of air quality]c. sub. pa]
Is the specific heat of humid air, and [U. sub. a]is the air-
Depending on the coil surface area, the lateral heat transfer coefficient between the wet air and the coil surface.
Removal of residual heat by condensation occurs only in the part of the coil, where the surface temperature is lower than the dew point of the air passing through it.
The latent heat transfer rate on Daan is expressed as d [q. sub. al]= -[dot. m. sub. a][h. sub. g]d[w. sub. a]=[K. sub. M][h. sub. fg]([w. sub. a]-[w. sub. s])dA, (2)where [h. sub. fg]
It\'s steam heat ,[h. sub. g]
Is the specific enthalpy of saturated water vapor, and [K. sub. M]
According to the coil surface area, it is the transfer coefficient between the wet air and the coil surface.
Ignoring the enthalpy of condensed water, the steam heat is equal to the specific enthalpy of saturated water vapor. The water-
The edge thermal gain is expressed as d [q. sub. w]= [dot. m. sub. w][c. sub. pw]d[t. sub. w]=[U. sub. w]([t. sub. s]-[t. sub. w])dA, (3)where [dot. m. sub. w]is the chilled-
Water flow]c. sub. pw]is the chilled-
Specific heat of water and [U. sub. w]is the water-
Depending on the coil surface area, the side heat transfer coefficient between the coil surface and the chilled water.
The saturation humidity ratio of the coil surface is a function based on the saturation humidity ratio and surface temperature
Temperature curve given by ASHRAE (2001). [w. sub. s]= [w. sub. s]([t. sub. s])(4)The water-
If the enthalpy of condensate removal is ignored, the side heat gain should be balanced with the sum of total heat transfer, apparent heat transfer and latent heat transfer on the air side. d[q. sub. w]= d[q. sub. as]+ d[p. sub. al](5)
Equation 1 to 5 is the basic differential equation, which fully describes the heat transfer and heat transfer process of the cooling coil.
Five unknown variables-air dry-
Bulb Temperature ([t. sub. a])
Air humidity ratio ([w. sub. a])
Surface temperature of coil ([t. sub. s])
, Coilsurface humidity ratio ([w. sub. s])
Water temperature ([t. sub. w])--
It can be obtained by solving the above five equations.
Unfortunately, due to coupling and nonlinear heat-mass transfer processes, it is difficult to solve the differential equation on the entire coil.
For a small cooling coil, however, it is reasonable to assume that the apparent heat ratio is constant.
Therefore, heat transfer and mass transfer can be decoupled by using a constant sensible heat ratio. Then d[q. sub. w]= [a. sub. e]d[q. sub. as]= [b. sub. e]d[q. sub. al], (6)where SH[R. sub. e]= [q. sub. as]/[q. sub. w]= [[c. sub. pa]([t. sub. a,e]-[t. sub. a,l])]/[[h. sub. g]([w. sub. a,e]-[w. sub. a,l])+[c. sub. pa]([t. sub. a,e]-[t. sub. a,l])], [a. sub. e]= 1/SH[R. sub. e], and [b. sub. e]= 1/[1 -SH[R. sub. e]].
By substituting equation 1 into Equation 6, the coupled heat transfer equation can be obtained. d[q. sub. w]= [a. sub. e]d[q. sub. as]=-([a. sub. e][dot. m. sub. a])[c. sub. pa]d[t. sub. a]=([a. sub. e][U. sub. a])([t. sub. a]-[t. sub. s])dA (7)
The equivalent mass flow rate and heat transfer coefficient of the decoupling sensible heat transfer can be defined :[dot. m\'. sub. a]= [a. sub. e][dot. m. sub. a](8a)[U\'. sub. a]= [a. sub. e][U. sub. a](8b)[dot. m\'. sub. w]= [dot. m. sub. w](8c)[U\'. sub. w]= [U. sub. w](8d)
The differential equations of heat transfer for decoupling are derived from Equation 3 and equation 7. d[q. sub. w]= -[dot. m\'. sub. a][c. sub. pa]d[t. sub. a]=[U\'. sub. a]([t. sub. a]-[t. sub. s])dA (9)d[q. sub. w]= [dot. m\'. sub. w][c. sub. pw]d[t. sub. w]=[U\'. sub. w]([t. sub. s]-[t. sub. w])dA (10)
Decoupled differential equations 9 and 10 describe the sensitive heat transfer process in a cooling coil driven by the temperature difference of the assumed constant shrs.
Because these equations have the standard format of the heat exchanger analogy theory, the validity is used-NTU method (seeAppendix).
Similarly, the decoupling heat transfer differential equation on the air side can be obtained by substituting equation 2 into Equation 6. d[q. sub. w]= -([b. sub. e][dot. m. sub. a][[h. sub. g]/[c. sub. pa]])[c. sub. pa]d[w. sub. a]= ([b. sub. e][K. sub. M][h. sub. fg])([w. sub. a]-[w. sub. s])dA(11)
However, the driving force on the air side of equation 11 is the difference in humidity ratio ,[w. sub. a]-[w. sub. s](
Please note that for wet coils ,[w. sub. s]
The humidity ratio corresponding to the surface of the shrink water on the coil, rather than the humidity ratio of the underwater physical surface)
, The driving force in equation 3 on the water side is the temperature difference ,[t. sub. s]-[t. sub. w].
In order to establish the standard differential equation of the heat exchanger, the temperature difference in equation 3 must be converted to the humidity ratio difference.
Equation 4 relates the saturation temperature to the saturation humidity ratio, so the coil surface temperature [t. sub. s]
On the water side of the oil it is easy to convert to the virtual surface saturation humidity ratio corresponding [t. sub. s].
Along a small part of the oil, it is reasonable to consider that the surface temperature difference between the inlet and outlet of the element is a temperature difference, and at the entry and exit temperature nodes of the element, approximate the nonlinear saturation curve to a straight cut line on the saturation curve.
The slope of this cut line can be expressed as tan [[alpha]. sub. e]= [[w. sub. s,e]-[w. sub. s,l]]/[[t. sub. s,e]-[t. sub. s,l]], (12)where [[alpha]. sub. e]
Angle between the slope of the saturation curve at the element and the horizontal temperature axis ,[t. sub. s,e]and [t. sub. s,l]
Is the surface temperature of the entry and exit nodes, and [w. sub. s,e]and [w. sub. s,l]
Is the corresponding virtual surface humidity ratio determined [t. sub. s,e]and [t. sub. s,l]
Use equation 4.
Finally, the imaginary humidity ratio [w. sub. s]
The function of temperature in the element can be expressed [w. sub. s]= ([t. sub. s]-[t. sub. s,e])tan [[alpha]. sub. e]+[w. sub. s,e]. (13)
Similarly, the humidity ratio of the chilled water has no physical meaning, and a virtual value can be specified.
For example, labeling frozen foods is traditional
The state of water on the saturation curve in the psymetric chart.
In this case, according to the virtual drying ratio of the chilled water obtained
Use the water temperature of equation 4.
Of course, equation 13 can also be used to define another humidity ratio of chilled water, virtual chilled water
Water humidity ratio ([w. sub. w])
According to the change of water temperature ([t. sub. w])expressed as [w. sub. w]= ([t. sub. w]-[t. sub. s,e])tan [[alpha]. sub. e]+[w. sub. s,e]. (14)
Surface humidity ratio and virtual freezing with virtual coil-
Water humidity ratio, driving force of water
Side heat transfer in equation 3 is converted from temperature difference to humidity ratio difference, which is compatible with the driving force of the air
Side heat transfer in equation 11.
Substitute Equations 13 and 14 into equations 3, d [q. sub. w]becomes d[q. sub. w]= ([dot. m. sub. w][1/tan[[alpha]. sub. e]])[c. sub. pw]d[w. sub. w]=([U. sub. w][1/tan[[alpha]. sub. e]])([w. sub. s]-[w. sub. w])dA. (15)
The equivalent mass flow rate and heat transfer coefficient of decoupling heat transfer can be defined :[dot. m\". sub. a]= [b. sub. e][dot. m. sub. a][[h. sub. g]/[c. sub. pa]](16a)[U\". sub. a]= [b. sub. e][K. sub. M][h. sub. fg](16b)[dot. m\". sub. w]= [dot. m. sub. w][1/tan[[alpha]. sub. e]](16c)[U\". sub. w]= [U. sub. w][1/tan[[alpha]. sub. e]](16d)
Decoupling heat transfer differential equations are derived from equations 11 and 15. d[q. sub. w]= -[dot. m\". sub. a][c. sub. pa]d[w. sub. a]=[U\". sub. a]([w. sub. a]-[w. sub. s])dA (17)d[q. sub. w]= -[dot. m\". sub. w][c. sub. pw]d[w. sub. w]=[U\". sub. w]([w. sub. s]-[w. sub. w])dA (18)
Decoupled differential equations 17 and 18 describe the potential heat transfer process in a cooling coil driven by a humidity ratio difference, with two parameters assumed to be constant: the slope of the ratio of SHR to saturated humidity
Temperature curve.
Again, the element equation can be easily obtained using the validity
NTU method of heat exchanger analogy theory (see Appendix).
Because each element is treated as a small cross.
Flow heat exchanger, effectiveness of cross
Flow heat exchangers must be applied to develop the equation of elements described in the appendix.
Validity of crossover
ByIncropera and DeWitt give the flow heat exchanger (2002). [epsilon]= 1 -exp[[[NTU. sup. 0. 22]/[C. sub. r]]{exp[-[C. sub. r](NTU)[. sup. 0. 78]]-1}](19)
In the cooling coil simulation, the heat transfer area and coefficient of the simulation program on the air and water sides are considered as given inputs.
Leave the water temperature and dry air-
The bulb temperature and humidity ratio needs to be simulated according to the given water supply temperature and mass flow rate as well as the air supply temperature, humidity ratio and mass flow rate.
For the sharp slope of the saturated humidity ratio on the surface of the given coil, it is easy to simulate the performance of the cooling coil
Temperature of use effectiveness-
The NTU method and the finite element method, as shown in the appendix.
However, before the simulation, the SHRS and slope of the saturation humidity ratio on the surface of the coil at the single element operating point were unknown parameters.
Therefore, trial and error must be used in simulations using the following procedures. 1.
According to the number of lines of the coil, the cooling coil is discrete into many elements. 2.
Ratio of saturation humidity on the surface of the initial SHRS and coils
In the first Test, the slope value of the temperature curve of all finite elements can be assumed.
The initial value of SHRs is selected between 0. 5 and0.
9, and calculate the initial value of the slope of the saturation curve based on the water supply temperature and the water supply air drying
Use the bulbtemp feature of equation 4 and 12. 3.
The sensible heat and latent heat transfer modes are decoupled using equations 9, 10, 17 and 18. 4.
Calculate the node temperature and humidity ratio on both sides of air and water using effectiveness-
NTU methods and finite element methods listed in the appendix. 5.
Using Equation 6, you can update the SHRS for each element based on the calculated airnode condition. 6.
Coil surface temperature can be calculated using equations 9 and 10 based on the heat balance between the air and the water side.
Then, the saturation humidity ratio of the surface of the linear coil
Using equations 12 and 13, update the temperature curve in each element based on the calculated coil surface temperature. 7.
Repeat the simulation until all of the changes in both SHRS and slopes are very small.
Figure 2 shows the flow chart of the testand-error process.
Application and results to 10-
Drain the cooling coil.
The masstransfer coefficient is calculated with Lewis number 1.
The entity coil is divided into 10 elements, as shown in Figure 1b.
There is air in the cooling coil-
12,000 side UA value of Btu/h x [degrees]
F. at a airflow rate of 5000 cfm and water-
60,000 side UA value of Btu/h x [degrees]
F at a flow rate of 40 fpm. Thechilled-
Water supply temperature is 42 [degrees]
F. The temperature of the air entering is 86 [degrees]F. Three wet-
[Bulb conditions] 70degrees]F,67[degrees]F, and 61[degrees]
Select F to represent the process of complete wet, partial wet, and full dry cooling.
Flow rate is5000 cfm and adjust the flow rate to maintain the discharge air temperature of 55 [degrees]F.
As a result, the required flow rates are 40, 30 and 17.
5 gpm for these three conditions.
Figure 3 shows the performance of the simulated cooling coil in these three cases.
Figure 3a, 3c, and 3e map the cooling process using air and freezing equipment
The water node temperature and humidity ratio on the psymetric chart, as well as the average temperature and humidity ratio of the coil surface elements.
Figure 3b, 3d, and 3f give the distribution of the slope of the element SHRs and saturation curves along the coil under these three conditions.
As shown in Figures 3a and 3b, the coil surface humidity ratio (or dew-
Point temperature)
Lower than the air humidity ratio (or dew-
Point temperature)
For all nodes of the entire coil.
Element SHR changes from 0.
85 coil empty letter to 0.
Coil vent 54. [
Figure 2:
As shown in Figure 3c and 3d, in the condition of partial humidity, the saturation humidity ratio of the coil surface is higher than the air humidity ratio of the node closest to the air inlet.
Therefore, the surface of the oil is dry on these nodes.
There is no change in the air humidity ratio, and in this area the element SHRS remain uniform.
On the other hand, due to the low surface temperature of the coil, the rest of the coil area still has the ability to Dewet and the shrvalue of the element is less than 1. [
Figure 3 slightly]
As shown in figures 3e and 3f, the saturation humidity on the surface of the coil is higher than the air humidity ratio of all nodes in the whole coil under completely dry conditions.
Therefore, the whole coil is dry for all nodes, and the SHRS are 1. [
Figure 4 slightly]
In general, the element condition is described by a separate SHRvalue.
Therefore, the use of a general model with the slope of the variable element SHRs and the variable element saturation curve accurately simulates the cooling coil performance, which is automatically updated according to the actual coil condition.
For partial wet coil applications, track the element SHRs in each trial to evaluate the effect of the initial SHRs value on the simulated cooling process.
Figure 4 compares the elements SHRs in four different trials with an initial SHRs value of 0. 5 and 0. 9.
After the tenth trial, the SHRdistribution curve in both cases finally converge to the same curve.
This means that the initial SHR value does not affect the cooling process of the simulation.
The influence of the number of elements per row the air conditioner uses a different number of elements in each row for simulation.
One model uses one element for each row, as shown in Figure 1b, and the other model uses two elements for each row.
The simulation is in 10-
Coil and 4-row coil.
Figure 5 and 10-of the simulation-
The line coil of one and two elements per row, Figure 5b compares 4-row coil.
In both cases, both elements in each row have no significant effect on simulated air conditions, especially for 10-row coil.
This shows that onerowone-
The component coil separation method can achieve sufficient accuracy in many aspects, and the calculation time is shorter.
Conclusion The Heat ratio of individualsensible is utilized by the development of decoupling cooling coil model (SHR)
The value of each element of the coil.
The element equations of explicit heat transfer and latent heat transfer have the standard format of the analogy theory of heat exchanger, which can be solved with effectiveness
NTU method for all coil conditions.
The cooling process can then be easily determined using the finite element method.
The variable element SHR determines the condition of the coil, whether it is completely dry, completely wet or partially wet.
The model takes into account the actual SHRS and coil surface saturation temperature curves, which makes it more accurate than those that assume constant SHRS in oil.
Initial SHRS in the trial-and-
Error handling does not affect the simulation results.
At the same timerow one-
The component coil separation method realizes the simulation results equivalent to two components per line. [
Figure 5 Slightly]
Future work will be carried out on the cooling coil to test the theoretical model developed.
The results of these experiments will be presented in subsequent papers.
Named A = coil surface area ,[ft. sup. 2]([m. sup. 2])
Part A: application 32 (1):63-83. Elmahdy, A. H. , and G. P. Mitalas. 1977.
A simple model for calculating cooling and dehumidifying coils for building energy requirements.
Heating 83 (2):103-117. Incropera, F. P. , and D. P. DeWitt. 2002.
Introduction to heat transfer.
John Willie and his son. Khan, A. Y. 1994.
Analysis of heat transfer and mass transfer performance of partial cooling coils
Load running conditions.
ASHRAE transaction100 (1):54-62. McQuiston, F. C. 1975.
Fin efficiency with combined heat and masstransfer.
Heating 81 (1):350-355. McQuiston, F. C. 1978.
Heat, mass and momentum transfer data for five platesfin-
Tube heat transfer surface.
ASHRAE Transactions84 (1):266-293. Mirth, D. R. , and S. Ramadhyani. 1993.
Comparison of air simulation methods
Side heat transfer and mass transfer in cold-medium
Water cooling coil.
Heating 99 (2):285-299.
Appendix validity-
NTU method and finite element method differential equation on water side of heat exchanger: d [q. sub. w]= [dot. m. sub. w][c. sub. pw]d[t. sub. w]=[U. sub. w]([t. sub. s]-[t. sub. s])dA (A1)Air side: d[q. sub. a]= -[dot. m. sub. a][c. sub. pa]d[t. sub. a]=[U. sub. a]([t. sub. a]-[t. sub. s])dA (A2)
Heat balance: dq = d [q. sub. w]= d[q. sub. a](A3)
Heat transfer based on effectivenessNTU method U = [[U. sub. a][U. sub. w]]/[[U. sub. a]+ [U. sub. w]](A4)[C. sub. a]= [dot. m. sub. a][c. sub. pa](A5)[C. sub. w]= [dot. m. sub. w][c. sub. pw](A6)[C. sub. min]= min([C. sub. a], [C. sub. w])(A7)[C. sub. max]= min([C. sub. a], [C. sub. w])(A8)[C. sub. r]= [C. sub. min]/[C. sub. max](A9)NTU = UA/[C. sub. min](A10)[epsilon]= f(NTU, [C. sub. r])
Stream-based mode (A11)[q. sub. t]= [epsilon][C. sub. min]([t. sub. a,i]-[t. sub. w,i])(A12)
The element equation mass flow rate of the finite element method may change on both sides of the air and water inside the heat exchanger.
Along the element select the component mass flow as the average, the node mass flow can be used as the average mass flow of the two components connected by the node.
Figure A1 shows a schematic diagram of an element and its node parameters.
The node heat flow can be expressed as a function of the node temperature inside the element. [q. sub. a,i. sup. e]= -[dot. m. sub. a,i][c. sub. pa][t. sub. a,i. sup. e](A13)[q. sub. a,o. sup. e]= [dot. m. sub. a,o][c. sub. pa][t. sub. a,i. sup. e]-[epsilon][C. sub. min]([t. sub. a,i. sup. e]-[t. sub. w,i. sup. e])(A14)[q. sub. w,o. sup. e]= [dot. m. sub. w,o][c. sub. pw][t. sub. w,i. sup. e]-[epsilon][C. sub. min]([t. sub. a,i. sup. e]-[t. sub. w,i. sup. e])(A15)[
Figure A1 omitted][q. sub. w,i. sup. e]= -[dot. m. sub. w,i][c. sub. pw][t. sub. w,i. sup. e](A16)
Typically, these basic element equations can be written in a matrix format. [
Mathematical expressions that cannot be reproduced in ASCII](A17)
Integral the equation on the entire coil. The total heat flow to the node is contributed by all the elements it connects.
Finally, the heat flow equation of integral nodes can be obtained by superposition method. [
Mathematical expressions that cannot be reproduced in ASCII](A18)
Boundary conditions, since there is no net heat flow flowing in or out of the internal nodes, the heat flow can be set to zero.
However, the inlet and outlet nodes of the coil have different boundary conditions. Entering Nodes.
For water and air entering the node, the temperature ([t. sub. a,i]and [t. sub. w,i])are given.
Assume that the number of nodes entered in the product function is m.
In order to force a constant temperature on this node, the following changes can be made to the integral equation. [k. sub. mm]= [infinity](A19)[
Mathematical expressions that cannot be reproduced in ASCII](A20)Leaving Nodes.
For water and air leaving nodes, heat balance is applied to determine the temperature of leaving nodes.
Assume that the number of left nodes in the integral equation is n. [
Mathematical expressions that cannot be reproduced in ASCII](A21)
Because the temperature leaving the node is unknown, the heat balance is met by changing the matrix and setting the heat flow to zero. [q. sub. n]= 0 (A22)[
Mathematical expressions that cannot be reproduced in ASCII](A23)
Wang Gang, Dr. Liu Mingsheng, sports commissioner Ashley David E.
Dr. Claridge, sports commissioner ASHRAE Wang is an associate professor of research and Liu Mingsheng is a professor at the University of NE. Lincoln, Omaha. David E.
Claridge is a professor at the college station of Texas a & M University.
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