Heat Exchanger Design

Transcription

1 Heat Exchanger Design Heat Exchanger Design Methodology Design is an activity aimed at providing complete descriptions of an engineering system, part of a system, or just a single system component. These descriptions represent an unambiguous specification of the system/component structure, size, and performance, as well as other characteristics important for subsequent manufacturing and utilization. Design can be accomplished using a well defined design methodology. In a broad sense, the design of a new heat exchanger means the determination/selection of an exchanger construction type, flow arrangement, tube/plate and fin material, and the physical size of an exchanger to meet the specified heat transfer and pressure drops within all specified constraints. 1

2 Heat Exchanger Design From the quantitative analysis point of view, there are a number of heat exchanger design problems. Two of the simplest and most important problems are referred to as the rating and sizing problems. Rating Problem A rating problem, also sometimes referred to as the performance or simulation problem, is the determination of heat transfer and pressure drop performance of either an existing exchanger or an already sized exchanger to check vendor s design. Inputs to the rating problem are the heat exchanger construction, flow arrangement and overall dimensions, complete details on the materials and surface geometries on both sides including their non-dimensional heat transfer and pressure drop characteristics (Nu and f vs. Re), fluid flow rates, inlet temperatures, and fouling factors. 2

3 Heat Exchanger Design The fluid outlet temperatures, total heat transfer rate, and pressure drops on each side of the exchanger are then determined in the rating problem. Sizing Problem A sizing problem involves the determination/selection of exchanger construction type, flow arrangement, tube/plate and fin material, and/or the physical size (length, width, height, and surface areas on each side) of an exchanger to meet the specified heat transfer and pressure drops within specified constraints. 3

4 Heat Exchanger Design For a shell-and-tube exchanger, a sizing problem, in general, refers to the determination of shell type, diameter and length, tube diameter and number, tube layout, pass arrangement, and so on. For a plate exchanger, a sizing problem means the selection of plate type and size, number of plates, pass arrangements, gasket type, etc. Inputs to the sizing problem are surface geometries including their dimensionless heat transfer and pressure drop characteristics, fluid flow rates, inlet and outlet fluid temperatures, fouling factors, and pressure drops on each side. 4

5 Thermodynamic analysis and operating aspects are equally or sometimes more important for the design of heat exchangers. Heat exchanger theory, correlations, and geometrical properties are used to present the solution procedures for rating and sizing. Heat exchanger design methodology has a very complex structure. Moreover, a design methodology for a heat exchanger as a component must be consistent with the life-cycle design of the system. Lifecycle design assumes considerations organized in the following stages: - 1.Problem formulation (including interaction with a consumer). 5

6 2.Concept development (selection of workable designs, preliminary design). 3.Detailed design (design calculations and other pertinent considerations). 4.Manufacturing. 5.Utilization considerations (operation, phase-out, disposal). At the initial stage, an engineer must clearly formulate the problem (i.e. specify requirements and define the main goal of the system design based on a good understanding of customer needs), evaluates alternative concepts of the system design and selects one or more workable design solutions. 6

7 Based on this, the engineer has to complete a detailed sizing, costing, and optimization and propose a design solution. Simultaneously, project engineering (construction/manufacturing) considerations should be taken into account. The issues related to start-ups, transients, steady and erratic operations, and ultimately retirement should be considered as well. Through consideration of these steps, a design team reconsiders the conclusions and in the light of the constraints imposed, iterates one or more steps until all the requirements are met within the tolerable limits. Within the framework of these activities, a particular design methodology has to be developed. 7

8 Fig. 1 Heat exchanger design methodology 8

9 Fig. 1 illustrates a methodology for designing a new heat exchanger presented by Kays and London (1998), Taborek (1988), and Shah (1982) for compact and shelland-tube exchangers. This design procedure may be characterized as a case study method. Major design considerations include: - 1.Process and design specifications. 2.Thermal and hydraulic design. 3.Mechanical design. 4.Manufacturing considerations and cost. 5.Trade-off factors and system-based optimization. 9

10 These design considerations are usually not sequential; there could be strong interactions and feedback among the aforementioned considerations as indicated by double sided arrows in Fig. 1 and may require a number of iterations before the design is finalized. The overall design methodology is quite complex because of the many qualitative judgments in addition to quantitative calculations that must be introduced. Depending on the specific application, some but not necessarily all of the foregoing considerations of heat exchanger designs are applied in various levels of detail during the design process. 10

11 Process and Design Specifications The process and problem specification is one of the most important steps in heat exchanger design. A heat exchanger design engineer can add the most value by working together with a system design engineer to develop smart specifications for the heat exchanger that define an optimum system. The smart specifications must be completed based on discussions with the customer, on industry and customer standards, and on design engineer s own experiences. Process or design specifications include all necessary information to design and optimize a heat exchanger for a specific application. 11

12 It includes problem specifications for operating conditions, exchanger type, flow arrangement, materials, and design/manufacturing/operation considerations. In addition, the heat exchanger design engineer provides necessary and missing information on the minimum input specifications required. Problem Specifications The first and most important consideration is to select the design basis (i.e. design conditions). Next comes an analysis of the performance at the design and off-design conditions. 12

13 The design basis would require the specification of operating conditions and the environment in which the heat exchanger is going to be operated (i.e. fluid mass flow rates including fluid types and their thermophysical properties, inlet temperatures and pressures of both fluid streams, required heat duty and maximum allowed pressure drops on both fluid sides, fluctuations in inlet temperatures and pressures due to variations in the process or environment parameters, corrosiveness and fluids fouling characteristics, and the operating environment (e.g. safety, corrosion/erosion, temperature level, and environmental impact points of view)). 13

14 In addition, information may be provided on overall size, weight, and other design constraints including cost, materials to be used, alternative heat exchanger types, and flow arrangements. If too many constraints are specified, there may not be a feasible design, and some compromises may be needed for a solution. The heat exchanger designer and system design engineer should work together at this stage to prepare the complete smart specifications for the problem. 14

15 Exchanger Specifications Based on the problem specifications and the design engineer s experience, the exchanger construction type and flow arrangement are first selected. Selection of the construction type depends on fluids used on each side of the exchanger (gas, liquid, or condensing/evaporating), operating pressures, temperatures, fouling and cleanability, fluids and material compatibility, corrosiveness of the fluids, how much leakage is permissible from one fluid to the other, available heat exchanger manufacturing technology, and cost. 15

16 The choice of a particular flow arrangement is dependent on the required exchanger effectiveness, exchanger construction type, upstream and downstream channelling, packaging envelope/footprint, allowable thermal stresses, and other criteria and design constraints. The orientation of the heat exchanger, the locations of the inlet and outlet pipes, and so on, may be dictated by system and/or available packaging/footprint space and channelling. Next, the core or surface geometry (e.g. shell type, number of passes, baffle geometry, etc.) and material are selected. 16

17 The core geometry is selected for a shell-and-tube exchanger; surface geometry is chosen for a plate, extended surface, or regenerative heat exchanger. There are several quantitative and qualitative criteria for surface selection. The qualitative criteria for surface selection are the operating temperature and pressure, the designer s experience and judgment, fouling, corrosion, erosion, fluid contamination, cost, availability of surfaces, manufacturability, maintenance requirements, reliability, and safety. 17

18 For shell-and-tube exchangers, the criteria for selecting core geometry/configuration are desired heat transfer performance within specified pressure drops, operating pressures and temperatures, thermal/pressure stresses, the effect of potential leaks on the process, corrosion characteristics of the fluids, fouling, cleanability, maintenance, minimal operational problems (vibrations, freeze-up, instability, etc.), and total installed cost. For shell-and-tube exchangers, the tube fluid is usually selected as the one having more fouling, high corrosiveness, high pressure, high temperature, increased hazard probability, high cost per unit mass, and/or low viscosity. 18

19 Maximum allowable pressure drop will also dictate which fluid will be selected for the tube side (highpressure fluid) and which for the shell side. For compact heat exchangers, there may be a choice of considering offset strip fin, louver fin, or other fin geometry. For each fin geometry selected, carry-out the thermal/hydraulic design and mechanical design. Example 1 A hydrocarbon gas has to be cooled in a chemical plant. A stream of a liquid hydrocarbon is available to be used as a coolant. The gas stream has to change its temperature from C to C. 19

20 The inlet temperature of the liquid stream is C. The required enthalpy change of the hot gas is smaller than 300 kw (with a small mass flow rate of an order of magnitude 0.01 kg/s). Both fluids are at relatively high pressures (i.e. the respective pressures are of an order of magnitude 10 MPa). Assume design specifications for the heat exchanger types listed are valid, is it possible using this incomplete set of process data, to offer an unambiguous selection of a feasible heat exchanger type that will be capable of performing the task? Consider shell-and-tube, doublepipe, and spiral plate heat exchanger types. 20

21 Solution Problem Data and Schematic An incomplete database is provided to support a process specification; a situation often encountered in practice. All the information regarding various heat exchanger types under consideration (shell-and-tube, double-pipe, and spiral plate heat exchanger) are available. Analysis and Discussion One possible approach to select a feasible heat exchanger type is first to eliminate the types characterized with specifications that conflict with the process conditions. 21

22 The first important fact to be considered is related to the operating temperature ranges and pressures. A study of various designs leads to the conclusion that a plate heat exchanger cannot be used because the allowable operating pressures and temperatures are both substantially lower than the process condition data imposed by the problem formulation. A spiral heat exchanger can operate at much higher temperatures up to C, but the pressure limitation is 2 MPa so, only two remaining types should be considered (i.e. only shell-and-tube and double-pipe heat exchangers are feasible candidates as both can easily sustain high pressures and temperatures). 22

23 Consequently, other criteria such as the required heat exchanger effectiveness, heat load, fluid characteristics (fouling and corrosion ability), cost, and others should be considered for the selection. For a relatively small heat load (i.e. smaller than 500 kw), a double pipe heat exchanger would be a cost-effective solution. Also, a multi tube double-pipe heat exchanger with or without fins should be considered for higher performance if cost considerations support. Finally, a decision should be made whether to use finned or plain tubes in the double pipe multi tube heat exchanger selected. 23

24 Due to the fact that the heat exchanger should accommodate both gas and liquid, the heat transfer conductance (ha) on the gas side will be low with low gas mass flow rate. Hence, employing fins on the gas side will yield a more compact unit with approximately balanced (ha) values on the gas and liquid sides. The tube fluid (liquid hydrocarbon) is more prone to fouling so a double pipe multi tube heat exchanger with finned tubes on the hydrocarbon gas side and liquid hydrocarbon on the tube side has to be suggested as a feasible workable design. 24

25 Heat Exchanger Thermal Design Basic Thermal and Hydraulic Design Methods Based on the number of variables associated with the analysis of a heat exchanger, dependent and independent dimensionless groups are formulated. The relationships between dimensionless groups are subsequently determined for different flow arrangements. Depending on the choice of dimensionless groups, several design methods including LMTD correction factor, ε-ntu, and other methods are being used. Inputs to the thermal and hydraulic procedures are the surface heat transfer and flow friction characteristics (also referred to as surface basic characteristics), geometrical properties, and thermo-physical properties of fluids in addition to the process/design specifications. 25

26 Surface Basic Characteristics Surface basic characteristics on each fluid side are presented as Nu and f vs. Re curves in dimensionless form and as the heat transfer coefficient h and pressure drop Δp vs. the fluid mass flow rate ṁ or fluid mass velocity G in dimensional form. Accurate and reliable surface basic characteristics are a key input for exchanger thermal and hydraulic design. Surface Geometrical Properties For heat transfer and pressure drop analyses, at least the following heat transfer surface geometrical properties are needed on each side of a two-fluid exchanger: - 26

27 1.Minimum free-flow area A o. 2.Core frontal area A fr. 3.Heat transfer surface area A which includes both primary and fin area if any. 4.Hydraulic diameter D h. 5.Flow length L. These quantities are computed from the basic dimensions of the core and heat transfer surface. On the shell side of a shell-and-tube heat exchanger, various leakage and bypass flow areas are also needed. 27

28 Thermo-physical Properties For thermal and hydraulic design, the following thermophysical properties are needed for the fluids: dynamic viscosity ν, density ρ, specific heat c p, and thermal conductivity k. For the wall, material thermal conductivity and specific heat may be needed. Thermal and Hydraulic Design Problem Solution Solution procedures for rating and sizing problems are of analytical or numerical nature, with empirical data for heat transfer and flow friction characteristics and other pertinent characteristics. 28

29 Due to the complexity of the calculations, solution procedures are often executed using commercial or proprietary computer codes. There are many geometrical and operating condition related variables and parameters associated with the sizing problem. Formulating the best possible design solution (i.e. selection of the values of these variables and parameters) among all feasible solutions that meet the performance and design criteria is achieved by employing mathematical optimization techniques after initial sizing to optimize the heat exchanger design objective function within the framework of imposed implicit and explicit constraints. 29

30 To analyze a exchanger heat transfer problem, a set of assumptions are introduced so that the resulting theoretical models are simple enough for the analysis. The following assumptions and/or idealizations are made for the exchanger heat transfer problem formulations (i.e. the energy balances, rate equations, boundary conditions, and subsequent Analysis): - 1.The heat exchanger operates under steady-state conditions (i.e. constant flow rates and fluid temperatures (at the inlet and within the exchanger) independent of time). 2.Heat losses to or from the surroundings are negligible (i.e. the heat exchanger outside walls are adiabatic). 30

31 3.There are no thermal energy sources or sinks in the exchanger walls or fluids, such as electric heating, chemical reaction, or nuclear processes. 4.The temperature of each fluid is uniform over every cross section in counterflow and parallel flow exchangers (i.e. perfect transverse mixing and no temperature gradient normal to the flow direction). Each fluid is considered mixed or unmixed from the temperature distribution viewpoint at every cross section in single-pass cross flow exchangers, depending on the specifications. For a multi pass exchanger, the fluid is considered mixed or unmixed between passes. 31

32 5.Wall thermal resistance is distributed uniformly in the entire exchanger. 6.Either there are no phase changes (condensation or vaporization) in the fluid streams flowing through the exchanger or the phase change occurs under constant temperature as for a single-component fluid at constant pressure; the effective specific heat for the phasechanging fluid is infinity, and hence C max = ṁc p,eff ; where ṁ is the fluid mass flow rate. 7.Longitudinal heat conduction in the fluids and in the wall is negligible. 32

33 8.The individual and overall heat transfer coefficients are constant (independent of temperature, time, and position) throughout the exchanger, including the case of phasechanging fluids. 9.The specific heat of each fluid is constant throughout the exchanger, so that the heat capacity rate on each side is treated as constant; the other fluid properties are not involved directly in the energy balance and rate equations but are involved implicitly in NTU and are treated as constant. 10.For an extended surface exchanger, the overall extended surface efficiency η o is considered uniform and constant. 33

34 11.The heat transfer surface area A is distributed uniformly on each fluid side in a single-pass or multi pass exchanger. In a multi pass unit, the heat transfer surface area is distributed uniformly in each pass, although different passes can have different surface areas. 12.For a plate-baffled 1 n shell-and-tube exchanger, the temperature rise (or drop) per baffle pass (or compartment) is small compared to the total temperature rise (or drop) of the shell fluid in the exchanger, so that the shell fluid can be treated as mixed at any cross section. This implies that the number of baffles is large in the exchanger. 34

35 13.The velocity and temperature at the entrance of the heat exchanger on each fluid side are uniform over the flow cross section. There is no gross flow maldistribution at the inlet. 14.The fluid flow rate is uniformly distributed through the exchanger on each fluid side in each pass; i.e. no passage-to-passage or viscosity-induced maldistribution occurs in the exchanger core. Also, no flow stratification, flow bypassing, or flow leakages occur in any stream. The flow condition is characterized by the bulk (or mean) velocity at any cross section. 35

36 Problem Formulation The objective of performing exchanger heat transfer analysis is to relate the heat transfer rate q, heat transfer surface area A, heat capacity rate C of each fluid, overall heat transfer coefficient U, and fluid terminal temperatures. Two basic relationships are used for this purpose: - 1.Energy balance based on the first law of thermodynamics. 2.Rate equations for heat transfer, as outlined by equations 1 and 2. 36

37 Fig. 3 Nomenclature for heat exchanger variables 37

38 Consider as an example, the two-fluid counter flow heat exchanger (Fig. 3) to arrive at the variables relating to the thermal performance of a two-fluid exchanger. Schematic of Fig. 3 and the balance equations for different exchanger flow arrangements may be different, but the basic concept of modelling remains the same. For steady state flow, overall adiabatic system, and negligible potential and kinetic energy changes, two differential energy conservation equations based on the energy balance implied by the first law of thermodynamics can be combined as follows for control volumes associated with the differential element of area da:- 38

39 Where; dq is the heat transfer rate from the hot to the cold fluid, C = ṁc p the heat capacity rate of the fluid, ṁ the fluid mass flow rate, c p the fluid specific heat at constant pressure, dt h = T h,o T h,i and dt c = T c,o T c,i are temperature ranges; the subscripts h and c denote hot and cold fluids, respectively. The negative signs in equation 1 are a result of T h and T c decreasing with increasing A (i.e. with increasing flow length). In general, for any isobaric change of state, equation 1 should be replaced by: - Where; h is the fluid specific enthalpy (J/kg). 39

40 If the change of state is a phase change, enthalpy differences should be replaced by enthalpies of phase change (either enthalpy of evaporation or enthalpy of condensation). However, c p can be assumed as infinity for condensing or evaporating single-component fluid streams. Hence, the phase changing stream can be treated as a single-phase fluid having ΔT = q/c or dt = dq/c, with C being infinity for a finite q or dq since the ΔT (ΔT = T h,i T h,o or T c,o - T c,i as appropriate) or dt = 0 for isothermal condensing or evaporating fluid streams. 40

41 The overall heat transfer rate equation on a differential base for the surface area da of Fig. 3 is: - Where; U is the local overall heat transfer coefficient; the local temperature difference ΔT = (T h - T c ), and the thermal conductance UdA is the driving potential for heat transfer. Integration of equations 1 and 3 together over the entire heat exchanger surface for specified inlet temperatures will result in the ε-ntu relation; an expression that relate all important operating variables and geometry parameters of the exchanger. The common assumptions and/or idealizations invoked for the derivation and integration of equations 1 and 3 are summarized before. 41

42 Two basic energy conservation and rate equations could also be written on an overall basis for the entire exchanger under the conditions implied by the above mentioned idealizations as follows: - Where; i and o denote inlet and outlet, respectively, T h,o and T c,o represent outlet temperatures (bulk temperatures T m for a non-uniform temperature distribution across a cross section), and U m and ΔT m the mean overall heat transfer coefficient and the exchanger mean temperature difference, respectively. 42

43 The rate equation 3, after rearrangement, is presented in integral form as: - Therefore, the mean temperature difference and mean overall heat transfer coefficient can be defined using the terms in equation 6 as: - 43

44 The fluid bulk temperature T m at an arbitrary duct cross section is defined as: - Where; u m is the fluid mean axial velocity defined as the integrated average axial velocity with respect to the freeflow area A o : - 44

45 From equations 4 and 5 and Fig. 3, the steady-state overalladiabatic heat exchanger behaviour can thus be presented in terms of dependent fluid outlet temperatures or heat transfer rate as functions of four operating condition variables and three designer controlled parameters: - T h,0, T c,0, or q T, T, C, C, UA, flow arrangement h,i c,i h c dependent variables operating conditions variables parameters under design control 11 independent variables and parameters Equation 11 represents a total of six independent and one or more dependent variables for a given heat exchanger flow arrangement. 45

46 Of course, any one of the independent variables/parameters in equation 11 can be made dependent (if unknown); in that case, one of the three dependent variables in equation 11 becomes an independent variable/parameter. Thus the most general heat exchanger design problem is to determine any two unknown variables from this set when the rest of them are known. As it is difficult to understand and work with such a large number of variables and parameters as outlined in equation 11, three dimensionless groups are formulated from six independent and one or more dependent variables of equation

47 The reduced number of non-dimensional variables and parameters simplifies much of the analysis, provides a clear understanding of performance behaviour, and the results can be presented in more compact graphical and tabular forms. The specific form of these groups is to some extent optional. Depending on which method of heat transfer analysis used, the effectiveness number of heat transfer units (ε-ntu) method and the logarithmic temperature difference (LMTD) method options have been used. 47

48 Considering seven variables of the heat exchanger design problem (equation 11), there are a total of 21 problems, as shown in Table 1. In this table, the dimensionless parameters are also included with known or unknown values based on the dimensional variables. If only one of the temperatures is known, the specific heat is evaluated at that temperature for the determination/estimate of C and hence R (ratio of heat capacity rate of one side to the other side); an iteration may be needed once both temperatures are known. If specific heats are treated as constants, C 1 and C 2 in the equations above are interchangeable with ṁ 1 and ṁ 2. 48

49 Table 1 Design Problems in Terms of Dimensional and Dimensionless Parameters 49

50 Also, the UA product is considered as one of the design parameters. In the foregoing count, q is not added since it can readily be calculated from the overall energy balance (equation 3.5) or the rate equation (equation 3.6 or 3.12). Alternatively, if q is given, one of the temperatures or the flow rates (heat capacity rate) could be unknown in Table 3. The first six problems can be considered as variations of the sizing problem and the next 15 problems as a variation of the rating problem. Using the ε-ntu method, these 21 problems can be solved as follows: - 50

51 1.If ε 1 and ε 2 and R 1 are known through the heat balance, NTU 1 can be calculated from the ε-ntu formula for a given flow arrangement, either straightforward or iteratively, depending on whether NTU can be expressed explicitly or implicitly. 2.If all four temperatures are known, both ε 1 and R 1 are known through their definitions and hence NTU 1 can be calculated for a given flow arrangement; then C 1 = UA/NTU 1 and C 2 = C 1 /R 1. Knowing the specific heats of the given fluids, one can determine ṁ 1 and ṁ 2. 51

52 3.If NTU 1 and R 1 are known, ε 1 can be determined using the appropriate formula. Knowing ε 1, R 1, and the definition of ε 1, the unknown temperatures can be computed. 4.For the iterative solution procedure, assume ṁ 1 or C 1 (ṁ 2 or C 2 ), Calculate R 2 = C 2 /C 1 (R 1 = C 1 /C 2 ). From the problem specifications, NTU 2 = UA/C 2 (NTU 1 = UA/C 1 ) and q = C 2 T 2,i T 2,o = C 2 T 1,i T 1,o are known. Hence, knowing NTU 2 (NTU 1 ) and R 2 (R 1 ) for a given flow arrangement, determine ε 2 (ε 1 ) using the appropriate formula. Subsequently, compute T 1,i (T 2,i ) from the definition of ε 2 (ε 1 ), and C 1 (C 2 ) from the energy balance C 1 (T 1,i T 1,o ) = q (C 2 (T 2,i T 2,o ) = q). 52

53 With this new value of C 1 (C 2 ), repeat all the calculations above. Continue to iterate until the successive values of C 1 (C 2 ) converge within the accuracy desired. If ε 2 (ε 1 ) and NTU 2 (NTU 1 ) are given. Compute R 2 (R 1 ) iteratively (using for example the Newton Raphson method) from the known ε- NTU formula, since it is an implicit function of ε 2 (ε 1 ) and NTU 2 (NTU 1 ). Problems 18 and 20 can only be solved iteratively, with the solution procedure for problem 18 as follows. Assume C 2 and hence determine R 1 = C 1 /C 2. Also, compute NTU 1 = UA/C 1 from the input. For a given flow arrangement, determine P 1 using the appropriate formula from Table 3.6. Subsequently, compute T 1,i from the definition of P 1. 53

54 Finally, calculate C 2 from the overall energy balance C 1 (T 1,i T 1,o ) = C 2 (T 2,o T 2,i ). With this new value of C 2, repeat all the calculations above. Continue to iterate until successive values of C 2 converge within the desired accuracy. The solution procedures for problem 20 is identical to that for problem 18 just described, starting with assuming C 1 and computing R 2 and NTU 2. The solution procedure for problem 19 is relatively straightforward. In this case, P 2 and NTU 1 are known. For the given flow arrangement, select the P1-NTU1 formula from Table 3.6 and replace P 1 by P 2 = R 1 [see Eq. (3.98)]. 54

55 The resulting equation has only one unknown, R 1, since NTU 1 and P 2 are known; and R 1 is implicit. It can be computed iteratively using, for example, the Newton Raphson method. Similarly, P 1 and NTU 2 are known in problem 21, and compute R 2 iteratively after replacing P 1 by P 2 R 2 in the appropriate P-NTU formula of Table 3.6. See the footnote of Table 3.6 for how to convert P 1 - NTU 1 -R 1 formulas into P 2 -NTU 2 -R 2 formulas. 55

56 Example 2 Consider a heat exchanger as a black box with two streams entering and subsequently leaving the exchanger without being mixed. Assume the validity of both equations 1 and 2. Also take into account that ṁ j Δh j = (ṁc p ) j ΔT j, where ΔT j = T j,i T j,o. Note that regardless of the actual definition used, ΔT m must be a function of terminal temperatures (T h,i, T h,o, T c,i, and T c,o ). With these quite general assumptions, answer the following two simple questions: - 56

57 a.how many variables of the seven used on the right-hand sides of equations 1 and 2 should minimally be known, and how many can stay initially unknown, to be able to determine all design variables involved? b.using the conclusion from question (a), determine how many different problems of sizing a heat exchanger (UA must be unknown) can be defined if the set of variables includes only the variables on the right-hand sides of equations 1 and 2 (i.e. (ṁc p ) j ΔT j, T j,i, T j,o with j = 1 or 2, and UA). Assume the heat exchanger is adiabatic, the enthalpy changes in enthalpy rate equations can be determined by (ṁc p ) j ΔT j, and the heat transfer rate can be determined by equation 2. 57

58 Solution Problem Data A heat exchanger is considered as a black box that changes the set of inlet temperatures T j,i of the two fluids with (ṁc p ) j ΔT j, j = 1, 2 into the set of their respective outlet temperatures T j,o ( j = 1; 2) through heat transfer characterized by equation 2 (i.e. by the magnitude of UA). So this problem have to deal with the following variables (ṁc p ) 1 ΔT 1, (ṁc p ) 2 ΔT 2, T 1,i, T 1,o, T 2,i, T 2,o, and UA. 58

59 Analysis a.the answer to the first question is trivial and can be devised by straightforward inspection of the three equations given by equations 3 and 4 whose left hand sides are equal to the same heat transfer rate. This means that these three equations can be reduced to two equalities by eliminating heat transfer rate q. For example: - Where; ΔT m = f(t 1,i ; T 1,o ; T 2,i ; T 2,o ). 59

60 So there are two relationships between seven variables. Using the two equations, only two unknowns can be determined. Consequently, the remaining five variables must be known. b.the answer to the second question can now be obtained by taking into account the fact that: - i.ua must be treated as an unknown in all the cases considered (a sizing problem). ii.only two variables can be considered as unknown since we have only two equations at our disposal. 60

61 Thus, the number of combinations among the seven variables is six (i.e. in each case the two variables will be unknown and the remaining five must be known). This constitutes the list of six types of different sizing problems given in Table 1. Discussion and Comments Among the six types of sizing problems, four have both heat capacity rates (i.e. the products (ṁc p ) j ) known and, in addition to UA, the unknown is one of the four terminal temperatures. The remaining two problem types presented in Table 1 have one of the two heat capacity rates unknown and the other heat capacity rate and all four temperatures known. 61

62 Table 1: Heat Exchanger Sizing Problem Types 62