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Thermohydraulic Simulation of Heat Exchanger Networks Viviane B. G. Tavares,† Eduardo M. Queiroz,‡ and Andre´ L. H. Costa*,† Rio de Janeiro State UniVersity (UERJ), Instituto de Quı´mica, Rua Sa˜o Francisco XaVier, 524, CEP 20550-900-Rio de Janeiro, RJ, Brazil, and Federal UniVersity of Rio de Janeiro (UFRJ), Escola de Quı´mica, CT, Bloco E, Ilha do Funda˜o, CEP 21949-900-Rio de Janeiro, RJ, Brazil
This paper presents a simulation scheme of heat exchanger networks considering thermal and hydraulic effects simultaneously. The determination of network temperatures is carried out together with the evaluation of flow rates and pressures along the network, considering head losses in heat exchangers and associated piping. The model encompasses two systems of equations: a network hydraulic model, composed by a nonlinear system of mass balances and fluid flow equations, and an energy model, represented by a linear system of energy balances and heat exchanger equations. The mathematical structure is based on a matrix representation of the network. The proposed model allows a more realistic analysis of heat exchanger networks where flow rates are constrained by limitations of hydraulic facilities. The utilization of the simulation scheme is illustrated by the analysis of a cooling water system. 1. Introduction Thermal efficiency has become a key issue in chemical process industries. The reasons for this concern can be found in a modern scenario composed by crescent energy prices, environmental pressures for the limitation of carbon emissions, and increased worldwide market competition. In this scenario, energy integration through heat exchanger networks is an important tool in the effort to diminish energy consumption in process plants. Heat exchanger networks (HENs) can be described as a set of heat exchangers, mixers, and splitters interconnected to provide heat transfer between hot and cold process streams, thus allowing a reduction in the demand of utilities. The study of HEN synthesis has been the goal of a large variety of papers in the last decades.1 Despite the importance of the synthesis problem, important challenges related to HEN operation demand more attention.2 In this context, HEN simulation algorithms can be applied in the analysis of several problems, such as, determination of heat exchanger cleaning schedules, identification of optimal operating policies, network debottlenecking, etc. The basic formulation in the literature for the steady-state simulation of a HEN consists in the determination of network temperatures based on an algebraic model composed by energy balances associated to heat exchanger equations.3–5 Network flow rates are usually evaluated through mass balance equations for a given set of specifications of inlet flow rates and stream splits. However, in several process plant problems, flow rates may be constrained by hydraulic facility limitations. The conventional HEN analysis applied to these systems may imply considerable prediction errors, since network flow rates cannot be determined only by simple mass balance relations. The accurate determination of the actual flow rates must involve the solution of the network hydraulic model. Recently, several papers have discussed a hydraulic analysis of the fouling impact on crude preheat trains.6–8 In this new context, this paper studies the thermohydraulic simulation of HENs, where the proposed model can determine
network temperatures together with flow rates and pressures along the system, considering head losses in heat exchangers and associated piping. The proposed model addresses this problem employing a matrix approach, using the work of Oliveira Filho et al. as a starting point.9 The rest of this paper is organized as follows. Section 2 presents the individual models of the network elements, section 3 describes the representation of the HEN structure using digraphs, section 4 presents the HEN model, section 5 discusses the simulation algorithm, section 6 illustrates the application of the proposed algorithm in the analysis of a cooling water system, and section 7 reports the final conclusions. 2. Network Element Models The basic elements of a HEN are heat exchangers, mixers, and splitters. With the objective of simultaneously taking into account thermal and hydraulic behaviors, the HEN model demands the inclusion of pipe sections, network connections, and flow machines (pumps/compressors) in the set of network elements. 2.1. Heat Exchangers. The heat exchanger model is composed by mass and energy balances, a heat transfer equation (effectiveness relation), and a flow equation. Mass Balance. Mass balances for hot and cold streams around a heat exchanger yield: mh,i - mh,o ) 0
(1)
mc,i - mc,o ) 0
(2)
where m is the stream mass flow rate, the subscripts h and c represent the hot and cold streams, and the subscripts i and o represent the heat exchanger inlet and outlet, respectively. Effectiveness Equation and Energy Balance. According the ε-NTU method,10 the effectiveness of a heat exchanger, defined as the ratio between exchanger heat load and the maximum heat load established by thermodynamic limits, can be expressed by one of the following equations:
* To whom correspondence should be addressed. E-mail address:
[email protected]. † Rio de Janeiro State University (UERJ). ‡ Federal University of Rio de Janeiro (UFRJ). 10.1021/ie901275z 2010 American Chemical Society Published on Web 04/26/2010
ε)
Th,i - Th,o Th,i - Tc,i
(3)
Ind. Eng. Chem. Res., Vol. 49, No. 10, 2010
ε)
Tc,o - Tc,i Th,i - Tc,i
(4)
where eq 3 is valid if the heat capacity flow rate of the cold stream is larger than the hot stream one; otherwise, eq 4 is applied. Complementarily, the energy balance yields Cc(Tc,o - Tc,i) ) Ch(Th,i - Th,o)
(5)
where C is the heat capacity flow rate. Equations 3-5 can be organized in a single linear structure:
[
ε + (y - 1) 1-y y-ε -y CR(y - 1) + y CR(1 - y) - y CRy + (y - 1) -CRy + (1 - y)
[]
]
Tc,i Tc,o Th,i Th,o
) 0_
(6)
where CR is the ratio between the heat capacity flow rates. If the heat capacity flow rate of the cold stream is larger than the hot stream one, then y ) 1; otherwise, y ) 0. Flow Equation. A mechanical energy balance in a heat exchanger corresponds to Pi - Po - FgF(m) + Fg(zi - zo) ) 0
(7)
where P is the stagnation pressure, g is the gravitational acceleration, F is the fluid density, z is the elevation, and the function F expresses the relationship between head loss and heat exchanger flow rate. For shell-and-tube heat exchangers, ignoring the nozzles, the head loss in the tube side can be evaluated by the Darcy-Weisbach equation summed with a term related to exchanger headers:11
[(
Ftube(m) ) f
) ]
Ltube V2 + Kh Npt (Dtube - 2δtube) 2g
(8)
where f is the Darcy friction factor,12 Ltube is the tube length, Dtube is the outer tube diameter, δtube is the tube thickness, V is the fluid velocity, Npt is the number of tube passes, and Kh is a coefficient equal to 0.9 for one tube-side pass and 1.6 for two or more tube-side passes. For shell side flow, the head loss can be determined based on the Bell-Delaware method. According to this method, the head loss is evaluated by the sum of three terms (excluding nozzles and impingement devices): Fshell ) Fshell,c + Fshell,w + Fshell,e
(9)
where Fshell,c is the contribution of pure cross-flow, Fshell,w is the contribution of the flow in the baffle windows, and Fshell,e corresponds to the flow in the end zones (first and last baffle spacing). A detailed description for the evaluation of each term in eq 9 can be found in the work of Taborek.13 2.2. Mixers. This kind of element relates two or more inlet streams to one outlet stream, according to mass and energy balances. The mixer equations are valid for hot and cold streams. Mass Balance. The mass conservation principle applied to a mixer gives mi,1 + mi,2 + mi,3 + . . . - mo ) 0
(10)
Energy Balance. The energy conservation principle yields Ci,1Ti,1 + Ci,2Ti,2 + Ci,3Ti,3 + . . . - CoTo ) 0
(11)
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2.3. Splitters. In this work, the splitter is represented by one inlet stream and two outlet streams (if necessary, a combination of splitters can divide an inlet stream into any desired number of outlet streams). The splitter model equations are valid for hot and cold streams. Mass Balance. The mass balance equation corresponds to mi - mo,1 - mo,2 ) 0
(12)
Energy Balance. In a similar form, the energy balance equation is CiTi - Co,1To,1 - Co,2To,2 ) 0
(13)
Additionally, eq 14 must be included in the splitter model to establish that there is no temperature modification between inlet and outlet streams: Ti - To,1 ) 0
(14)
2.4. Pipe Sections. The model of a pipe assumes incompressible and isothermal flow and is represented by the corresponding flow equation: Pi - Po - FgFpipe(m) + Fg(zi - zo) ) 0
(15)
Theheadlossfunction,Fpipe,correspondstotheDarcy-Weisbach equation for a single pipe: Fpipe(m) ) f
( )
Lpipe V2 Dpipe 2g
(16)
Head losses due to fittings can be included using the concept of equivalent length. 2.5. Flow Machines. The adopted model for flow machines (pumps and compressors) is based on a direct relation between developed head and volumetric flow rate. Thus, the flow equation is given by Pi - Po - FgFmachine(m) + Fg(zi - zo) ) 0
(17)
where the term (-Fmachine) corresponds to the characteristic curve, represented by a polynomial: -Fmachine )
∑aq
j
j
)
∑ a (m/F)
j
j
(18)
where q is the volumetric flow rate. Since, thermodynamic equations of fluid compression are not included in the proposed model, the relation between inlet and outlet temperatures is Ti - To ) 0
(19)
2.6. Network Connections. A network connection can be employed to link different network elements. Mass Balance. The mass balance equation corresponds to mi - m o ) 0
(20)
Energy Balance. In a similar form, the energy balance equation is CiTi - CoTo ) 0
(21)
3. Network Representation Graphs and digraphs are mathematical representations of the links (called edges) between a set of elements (called vertices). A digraph is a graph where edges have a defined orientation.
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Figure 1. Process flowsheet digraph of example 1: (cold streams) continuous lines; (hot streams) dashed lines; (supply units) white squares; (demand units) black squares.
These concepts are widely used in the analysis of several engineering systems.14 Examples of the use of graphs and digraphs for the solution of HEN synthesis problems can be found in the work of Zhu et al.,15 Mehta et al.,16 and Shivakumar and Narasimhan.17 The analysis of the heat transfer along a HEN involves the description of the network as a process flowsheet, while the analysis of the flow implies the description of the HEN as a pipe network. However, the proposed approach does not involve two independent information structures; the matrix representation of the corresponding pipe network is derived from the matrix representation of the HEN flowsheet. For the process flowsheet digraph, heat exchangers, mixers, splitters, and network connections are represented by vertices and the streams are represented by edges. In the corresponding pipe network digraph, the description is similar but the heat exchangers are represented by edges instead of vertices. This alteration must be done because the pipe network model describes the pressure changes due to fluid flow occurring along the edges with the pressures represented as vertex variables. 3.1. Process Flowsheet Representation. A HEN flowsheet is composed by N vertices, representing NP external units (NPS supply units and NPD demand units, describing the other plant equipment), NHE heat exchangers, NMX mixers, NSP splitters, and NCN connections. These vertices are interlinked by S edges, representing Sc cold streams and Sh hot streams. The network connectivity is described by the incidence matrix M b (dimension N × S). Each vertex t of the network is associated to a matrix row, and each edge k is associated to a matrix column. A matrix element (M b )t,k is equal to 1 if edge k is direct to vertex t, -1 if edge k is directed from vertex t, or 0 if edge k is not linked to vertex t. The identification of the vertices and edges in the HEN flowsheet is defined according to the following incidence matrix partition:
[ ][ ]
M bPc M bP M bHE M bHE c MX MX M b M b) M ) b c M bSP M bSP c M bCN M bCN c
M bPh
M bHE h
M bMX h
(22)
M bSP h M bCN h
Figures 1 and 2 present the digraphs of the flowsheet of two smalls HEN examples (edges are numbered with Arabic numerals and vertices are numbered with Roman numerals). Figures 3 and 4 depict the corresponding incidence matrix in each example (only nonzero entries are shown and dashed lines indicate the matrix partition).
Figure 2. Process flowsheet digraph of example 2: (cold streams) continuous lines; (hot streams) dashed lines; (supply units) white squares; (demand units) black squares; (splitter) white circle; (mixer) black circle; (network connection) gray circle; (flow machine) edge with pump representation.
Figure 3. Incidence matrix of the process flowsheet digraph of example 1.
Figure 4. Incidence matrix of the process flowsheet digraph of example 2.
3.2. Pipe Network Representation. In the generation of the pipe network digraph from the process flowsheet digraph, each vertex of a heat exchanger is substituted by two new edges (cold and hot) with four new terminal vertices. The new edges are identified according to the following order of the original elements: heat exchanger cold streams and heat exchanger hot streams. The identification of the new terminal vertices obeys the following order: inlet heat exchanger vertices of cold streams, inlet heat exchanger vertices of hot streams, outlet heat exchanger vertices of cold streams, and outlet heat exchanger vertices of hot streams. The original heat exchanger vertices are deleted. According to this pattern, the incidence matrix of the pipe network digraph, Kb (dimension (N + 3NHE) × (S + 2NHE)),
[ ]
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is derived from the incidence matrix of the flowsheet representation:
[]
K bP K bMX K b) K bSP ) K bCN K bHE
M bPc
M bPh
M bMX c M bSP c M bCN c (M bHE c )+
M bMX h M bSP h M bCN h 0b
0b (M bHE h )+ HE 0b (M bc )HE 0b (M bh )-
0b 0b
0b 0b
0b 0b
0b 0b
0 -Ib b 0 -Ib b I 0b b 0b
(23)
I b
where 0b is a matrix of zeros and Ib is an identity matrix. The subscript “-” corresponds to the negative elements of the matrix. The subscript “+” indicates the opposite. Figures 5 and 6 present the corresponding digraphs of the pipe network representation of the HEN examples. Figures 7 and 8 present the incidence matrix of each network. 4. Network Model The network model is composed by two systems of equations which are solved sequentially: a hydraulic model and an energy model. The network hydraulic model contains mass balance equations at vertices and flow equations along edges. The network energy model is composed by energy balances and heat exchanger equations. According to the nature of the heat transfer and fluid flow equations adopted to describe the behavior of the network elements, the proposed network model can be employed to simulate HENs with streams without phase change. However, the temperature-enthalpy profile of phase change streams may be linearized, thus allowing in some situations the extension of the model to represent networks with condensers or vaporizers.
Figure 5. Pipe network digraph of example 1: (cold streams) continuous lines; (hot streams) dashed lines; (supply units) white squares; (demand units) black squares; (triangles) modified vertices (the new edges and vertices are written in bold type and underlined).
The discussion of this extension in the context of HEN synthesis can be found in the work of Smith.18 4.1. Variables. Mass flow rates and temperatures of the network streams are organized in the vectors m _ and T _ (dimension S × 1), which can be partitioned in cold and hot stream subvectors: _ cT m _ hT ] m _ T ) [m
(24)
_cT T _hT ] T _T ) [T
(25)
In the network hydraulic model, the vector _k (dimension (S + 2NHE) × 1) is an extension of the mass flow rate vector, in order to accommodate the new edges originated from the heat exchangers: _ cT m _ hT m _ nT ] _kT ) [m
(26)
where the subscript n identifies the new edges.
Figure 6. Pipe network digraph of example 2: (cold streams) continuous lines; (hot streams) dashed lines; (supply units) white squares; (demand units) black squares; (splitter) white circle; (mixer) black circle; (network connection) gray circle; (triangles) modified vertices; (flow machine) edge with pump representation (the new edges and vertices are written in bold type and underlined).
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equations), mixers (NMX equations), splitters (NSP equations), connections (NCN equations), and heat exchangers (4NHE equations), respectively. It corresponds to the matrix representation of eqs 1, 2, 10, 12, and 20: K bP_k + n_ ) 0_
(30)
K bMX_k ) 0_
(31)
K bSP_k ) 0_
(32)
K bCN_k ) 0_
(33)
Figure 7. Incidence matrix of the pipe network digraph of example 1 (the new edges and vertices are written in bold type and underlined).
K bHE_k ) 0_
(34)
The pressures along the network vertices are described by the vector P _ (dimension (N + 3NHE) × 1) related to the pipe network digraph:
Flow Equations. This subset of equations contains the flow equations of heat exchangers (2NHE equations), pipe sections, and flow machines (S equations). It is the matrix representation of eqs 7, 15, and 17:
T P _ T ) [(P _ MX)T (P _ SP)T (P _ CN)T (P _n ) ] _ P)T (P
(27)
where the superscript n identifies the new vertices. Fluid flow enters and leaves the network at the vertices representing external units. For each external vertex is associated a network inlet/outlet stream (not explicitly shown in the digraph representation). The mass flow rates and temperatures of those streams are organized in the vectors n_ and V _ (dimension NP × 1), which can be partitioned in supply and demand units:
_ (k_) + FgK bTz ) 0_ K bTP + FgF
(35)
where the vector _z is composed by the elevations of the network vertices. Each ith component of the function F _ is given by Fi ) Fpipe Fi ) -Fmachine
for i ∈ Qpipe for i ∈ Qmachine
(36) (37)
_nT ) [(n_PS)T (n_PD)T ]
(28)
Fi ) Ftube
for i ∈ Qtube
(38)
V _ T ) [(V _ PS)T (V _ PD)T ]
(29)
Fi ) Fshell
for i ∈ Qshell
(39)
The network hydraulic model is employed to determine the values of the variable vectors _k, n_, and P _ . The network energy model involves the determination of the variable vectors T _ and V _ , where the mass flow rate vector m _ has a fixed value obtained from the solution of the vector _k. 4.2. Network Hydraulic Model. The network hydraulic model is composed by mass balances at network vertices and fluid flow equations (mechanical energy balances along the edges). Mass Balances at Network Vertices. This subset of equations contains the mass balances at the external units (NP
where Qpipe and Qmachine represent the sets of pipes and flow machines, respectively, Qtube and Qshell are the sets of heat exchangers, tube side and shell side, respectively. It is important to observe that the parameters of the functions Fpipe, Fmachine, Ftube, and Fshell assume different values for each network element. Specifications. For each vertex of an external unit, it is specified the pressure or the flow rate (NP equations):
[ ]
E b
P _ PS ) _e* n_PS
Figure 8. Incidence matrix of the pipe network digraph of example 2 (the new edges and vertices are written in bold type and underlined).
(40)
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where the element (t, i) of the matrix Eb (dimension N × 2N ) is equal to 1, if the specification associated to the vertex t corresponds to the variable present in the position i of the vector of the pressures and flow rates; otherwise, it is zero. The vector _e* (dimension NPS × 1) contains the values of the specifications. 4.3. Network Energy Model. The network energy model is developed based on the proposal of Oliveira Filho et al.9 Energy Balances at External Units. This balance is represented by a temperature equality condition (NPS + NPD equations): PS
PS
M bPST _+V _ PS ) 0_
(41)
M bPDT _-V _ PD ) 0_
(42)
Heat Exchanger Equations. The matrix representation of the heat exchanger model, derived from eq 6, is (2NHE equations) [(Eci b)(M bHE b)(M bHE _c + [(Ehi b)(M bHE c )+ - (Eco c )-]T h )+ HE (Eho b)(M bh )-]T _h ) 0_ (43) _c + [(Rh b)(M bHE _h ) 0_ [(Rc b)(M bHE c )]T h )]T
(44)
where the auxiliary matrices in eqs 43 and 44 are diagonal matrices with the ith element of the main diagonal defined by (Eci b)i ) εi + (yi - 1)
(45)
(Eco b)i ) 1 - yi
(46)
(Ehi b)i ) yi - εi
(47)
(Eho b)i ) -yi
(48)
(Rc b)i ) CRi(yi - 1) + yi
(49)
(Rh b)i ) CRiyi + (yi - 1)
(50)
Figure 9. Algorithm structure.
Energy Balances at Mixers, Splitters and Connections. The energy balances at mixers, splitters, and connections are obtained from eqs 11, 13 and 21 (NMX +NSP + NCN equations): M bMX[diag _ )]T _ ) 0_ b(C
(51)
M bSP[diag _ )]T _ ) 0_ b(C
(52)
M bCNT _ ) 0_
(53)
where C _ is the vector of heat capacity flow rates of network streams. Split Fraction Equations. These equations correspond to the auxiliary equations of the energy balances at the splitters, described in eq 14 (NSP equations): SP [M b+ - SP b]T _ ) 0_
(54)
b is an auxiliary matrix (dimension NSP × S) dewhere SP SP bthrough the following transformation: each fined from M SP row of M b- has two nonzero entries, and in the corresponding b, the first nonzero element is multiplied by -1 and row of SP the subsequent element is vanished. Specifications. The temperatures of the streams associated to supply units are specified (NPS equations): PS
V _
- (V _ )* ) 0_ PS
(55)
Figure 10. Base case flowsheet. Table 1. Thermal Tasks
heat exchanger
hot stream mass flow rate (kg/s)
hot stream inlet temperature (°C)
hot stream outlet temperature (°C)
cooling water supply (kg/s)
E-101 E-102 E-103
12.0 12.0 10.5
80 120 80
65 80 40
11.8 28.7 35.6
Table 2. Physical Properties of the Hot Streams heat exchanger
density (kg/m3)
viscosity (Pa · s)
specific heat (J/(kg K))
thermal conductivity (W/(m K))
E-101 E-102 E-103
800 850 750
2.0 × 10-3 1.0 × 10-2 3.4 × 10-4
2200 2000 2840
0.15 0.10 0.19
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Table 3. Design Data of the Heat Exchangers heat exchanger
surface area (m2)
number of tube passes
number of tubes
tube length (mm)
baffle spacing (mm)
baffle cut (%)
shell diameter (mm)
E-101 E-102 E-103
27.3 76.7 133.6
4 4 4
125 301 459
3658 4267 4877
152 147 177
25 25 25
337 489 591
Table 4. Design Performance of the Heat Exchangers heat exchanger
overdesign (%)
tube side film coefficient (W/(m2 K))
shell side film coefficient (W/(m2 K))
total fouling factor (m2 K/W)
dirty overall coefficient (W/(m2 K))
tube side ∆P (bar)
shell side ∆P (bar)
E-101 E-102 E-103
13.6 5.7 5.2
10615 10651 8864
658 266 923
0.00051 0.00076 0.00048
460 214 578
0.54 0.62 0.47
0.14 0.11 0.05
Table 5. Length and Diameter of the Pipe Sections pipe section
2
3
4
5
6
7
8
9
10
length (m) inner diameter (mm) nominal diameter (in.)
110 254.4 10
170 202.7 8
300 202.7 8
170 202.7 8
100 202.7 8
400 202.7 8
100 90.2 3 1/2
400 90.2 3 1/2
110 254.4 10
equation). These modifications results in a linear system of equations which can be solved directly. After the resolution of the hydraulic model, the next step consists in the solution of the linear system of the energy model. For a given set of mass flow rates already evaluated, the energy balances and heat exchanger equations are linear in relation to network temperatures (variable vectors T _ and V _ ). The dependence of the physical properties on the temperature is not explicitly addressed in the model, but it could be considered through an outer loop around the network energy model which would update the physical properties at each iteration.
Table 6. Simulation Results outlet temperature cooling flow rate temperature of variation from the water variation from heat the hot stream design value flow rate the design exchanger (°C) (°C) (kg/s) value (%) E-101 E-102 E-103
-0.4 -1.9 -1.8
64.6 78.1 38.2
-33.9 +18.1 +44.7
7.8 33.9 51.5
where (V _ PS)* is the vector of the specified values of supply temperatures. 5. Simulation Algorithm The simulation algorithm is illustrated by Figure 9. Initially, the problem parameters and specifications are organized in a matrix structure. Thus, the structure of the process flowsheet (represented by the incidence matrix M b ) is employed to create the structure of the corresponding pipe network (represented by the incidence matrix Kb ). Sincethepipenetworkstructureisdefined,theNewton-Raphson method is employed to solve the nonlinear system of equations of the hydraulic model. The resolution of this system allows the determination of the values of the pressures and flow rates along the network (variable vectors P _ , _k, and n_). An adequate initial estimate for this problem can be obtained by the solution of a linear system originated from the hydraulic model through a secant linearization of the pump curve and the utilization of the friction factor expression for laminar flow (Poiseuille
6. Results The application of the thermohydraulic simulation approach is illustrated through the analysis of a cooling water system, where the HEN must be able to supply this cold utility for a set of hot streams in order to accomplish process demands. The example is composed by two problems, involving the simulation of a typical network (base case) and its revamp (revamp case). 6.1. Base Case. The HEN responsible to distribute cooling water in an industrial unit is composed by three heat exchangers in parallel. This structure is illustrated by the flowsheet in Figure 10 (in order to make clearer the presentation of the layout, only the cold streams are numbered). The cooling tower was designed to supply cooling water at 32 °C with return at 40 °C. The thermal tasks which defined
Table 7. Additional Thermal Task heat exchanger
hot stream mass flow rate (kg/s)
hot stream inlet temperature (°C)
hot stream outlet temperature (°C)
cooling water supply (kg/s)
E-104
8.5
69
41
15.6
Table 8. Revamp 1: New Heat Exchanger Design Data heat exchanger
surface area (m2)
number of tube passes
number of tubes
tube length (m)
baffle spacing (mm)
baffle cut (%)
shell diameter (mm)
E-104
109.5
4
301
6096
147
25
489
Table 9. Revamp 1: Design Performance of the New Heat Exchanger heat exchanger
over design (%)
tube side film coefficient (W/(m2 K))
shell side film coefficient (W/(m2 K))
total fouling factor (m2 K/W)
dirty overall coefficient (W/(m2 K))
tube side ∆P (bar)
shell side ∆P (bar)
E-104
2.5
6187
450
0.00049
340
0.28
0.07
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Figure 11. Revamp 1 flowsheetsthicker lines: new pipe sections. Table 10. Revamp 1: Data of the New Pipe Sections pipe section
11
12
length (m) inner diameter (mm) nominal diameter (in)
350 90.2 3 1/2
350 90.2 3 1/2
the design of each heat exchanger are described in Table 1, and the physical properties of the hot streams are shown in Table 2. The equipment design data are presented in Table 3, where all heat exchangers were built using tubes of 19.05 mm outer diameter and 1.65 mm thickness (3/4 in BWG 16) on a 23.81 mm triangular pitch (30°, 15/16 in.). According to fouling concerns, the cooling water flows in the tube side of the heat exchangers. The allowable pressure drop adopted for both streams in the design is 0.7 bar.19 The performances of the heat exchangers at design conditions are shown in Table 4. According to the rating code employed, described in the work of Costa and Queiroz,20 the heat exchangers present a certain thermal overdesign,21 aiming to guarantee the desired heat load during the plant operation. Since, these heat exchangers were installed in the hydraulic circuit connected to the cooling tower, the simulation algorithm can be employed to describe the thermohydraulic behavior of the HEN. The length and diameter of each pipe section (schedule 40) is displayed in Table 5, with absolute roughness equal to 46 × 10-6 m. The pump is represented by its characteristic curve, with the following polynomial coefficients (see eq 18, with head in meters and volumetric flow rate in cubic meters per second): a0 ) 42 m, a1 ) -180.01 s/m2, and a2 ) -554.09 s2/m5. The pump suction pressure and the return pressure to the tower is 1 bar. The entire hydraulic installation is considered at the floor level, with the exception of the return point, located at 5 m high. The flow rates of the hot streams are kept at the
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design values. It is considered that the performance of the cooling tower is not disturbed by the return temperature of the water. The thermohydraulic simulation of the network provides the results presented in Table 6. The thermal results shows that all heat exchangers reach the cooling target, i.e., the outlet temperature of the hot streams are below the desired value. Analyzing the hydraulic results, the total cooling water flow rate (93.2 kg/s) is about 22% above the sum of the heat exchanger design values (76.1 kg/s). However, the cooling water distribution among the three exchangers presents a considerable unbalance, where heat exchanger E-101 receives much less cooling water than the design specification. Due to the overdesign, there is no violation of the temperature target in this exchanger. This analysis of the interrelation between thermal and hydraulic aspects of the HEN is an important potentiality of the proposed approach. 6.2. Revamp 1. In this scenario, a revamp of the process plant demanded the insertion of a new cooler in the existing system (E-104). The main structure of the remainder of the network is not modified. The new thermal service is described in Table 7. The physical properties of the new hot stream are identical to those related to heat exchanger E-101. It is important to observe that the necessary cooling water flow rate for exchanger E-104 is lower than the available operational capacity of the pump, considering the design demand of the other heat exchangers. The design data of this new heat exchanger and its design performance are presented in Tables 8 and 9, respectively. The layout of the flowsheet after the installation of exchanger E-104 is presented in Figure 11. The new splitter and mixer for the insertion of the new pipes were introduced in the original pipe sections 4 and 7, where the insertion points are located at 250 and 100 m along the pipes, respectively. The description of the new pipe sections, complementing the data for the thermohydraulic simulation is presented in Table 10. The analysis of the system with the new heat exchanger is presented in Table 11. The simulation results shows that the desired outlet temperature of the hot stream in heat exchanger E-104 is not reached. Although the exchanger design seems adequate (a positive overdesign) and the original operation point of the hydraulic facility presents a surplus of cooling water (17.1 kg/s, larger than E-104 demand), heat exchanger E-104 fails when it is installed in the cooling water facility. This poor performance occurs due to the low cooling water flow rate. Despite the design pressure drop of the exchanger is relatively low, the hydraulic behavior of the entire system is not able to supply an adequate flow rate. This important analysis could not be done through the evaluation of the design data of each heat exchanger separately or even through the utilization of a conventional HEN simulator, thus illustrating the importance of the thermohydraulic analysis.
Table 11. Revamp 1: Simulation Results heat exchanger
outlet temperature of the hot stream (°C)
temperature variation from the design value (°C)
cooling water flow rate (kg/s)
flow rate variation from the design value (%)
E-101 E-102 E-103 E-104
64.7 78.3 38.3 45.9
-0.3 -1.7 -1.7 +4.9
7.5 31.4 50.0 6.4
-36.4 +9.4 +40.4 -59.0
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Ind. Eng. Chem. Res., Vol. 49, No. 10, 2010
7. Conclusions This paper presents a thermohydraulic simulation algorithm for heat exchanger networks based on a matrix structure. The parametrization of the network layout in the mathematical model is conducted through the representation of the network by digraphs. The network model is composed by two systems of equations which are solved in sequence. The first system corresponds to mass balances and flow equations. Due to its nonlinear nature, this system is solved by the Newton-Raphson method. The second system represents the energy balances and heat transfer equations and does not present nonlinearities in relation to the unknown temperatures. The application of the simulation algorithm is illustrated by the analysis of two problems involving a cooling water system. In these problems, the performance of each heat exchanger in the network depends on the hydraulic results of the entire network. The analysis of this kind of system employing a conventional algorithm of heat exchanger network simulation, only composed by mass and energy balances would not be able to predict the real behavior of the process temperatures. Thus, for these situations, where the system flow rates are constrained by the hydraulic facilities, a thermohydraulic approach can be an important tool. Acknowledgment E.M.Q. thanks CNPq, and AL.H.C. thanks UERJ (Procieˆncia Program) for the financial support. All authors are also grateful from the financial support of Petrobras. List of Symbols a ) polynomial coefficients of the pump characteristic curve C ) heat capacity flow rate, W/°C CR ) ratio of heat capacity flow rates, dimensionless D ) diameter, m _e* ) vector of pressure and flow rate specified values E b ) matrix for specification selection Eci b ) auxiliary matrix (eq 45) Eco b ) auxiliary matrix (eq 46) Ehi b ) auxiliary matrix (eq 47) Eho b ) auxiliary matrix (eq 48) f ) Darcy friction factor, dimensionless F ) relation between head variation and flow rate in the network elements, m g ) gravity acceleration, m2/s Ib ) identity matrix _k ) extended vector of stream mass flow rates for the network hydraulic model, kg/s Kh ) head loss coefficient of the heads of a heat exchanger, dimensionless K b ) incidence matrix of the pipe network digraph L ) length, m m ) mass flow rate, kg/s m ) vector of network stream mass flow rates, kg/s M b ) incidence matrix of the process flowsheet digraph N ) number of vertices Ntp ) number of tubes per pass n_ ) vector of external stream mass flow rates, kg/s P ) pressure, Pa P _ ) vector of network pressures, Pa Q ) sets of network elements Rc b ) auxiliary matrix (eq 49) Rh b ) auxiliary matrix (eq 50)
S ) number of edges SP b ) split fraction matrix T ) temperature, °C T _ ) vector of network stream temperatures, °C V ) fluid velocity, m/s V _ ) vector of external stream temperatures, °C y) binary parameter related to heat capacity flow rates z) elevation, m Superscripts CN ) connection HE ) heat exchanger MX ) mixer P ) external unit PD ) external demand unit PS ) external supply unit SP ) splitter T ) transpose n ) new vertex introduced in the pipe network digraph * ) variable specification value Subscripts c ) cold stream h ) hot stream i ) inlet o ) outlet k ) edge index t ) vertex index n ) new edge introduced in the pipe network digraph machine ) flow machine variable pipe ) pipe variable tube ) heat exchanger tube variable shell ) heat exchanger shell variable + ) matrix of positive entries - ) matrix of negative entries Greek letters δ ) tube thickness, m ε ) heat exchanger effectiveness, dimensionless F ) fluid density, kg/m3
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ReceiVed for reView August 14, 2009 ReVised manuscript receiVed March 11, 2010 Accepted March 25, 2010 IE901275Z
