Heat exchanger networks (HEN) connect hot and cold process streams and provide area over which heat transfer can occur. By facilitating heat transfer from process hot streams to process cold streams, the utility requirements of a system can be reduced. Retrofitting is the process of modifying a system or component after the original design. In the context of HEN retrofitting involves adjusting the system sequence, area, process conditions, and equipment. It is described as being more important and complex than HEN synthesis (Zhang & Rangaiah, 2013). Within this proposal the motivation for retrofitting and approaches available will be discussed and analysed. In addition, the outline for the project will be given with a time scaled plan of the work to be completed.
Produce a systematic methodology for the implementation of plate heat exchangers during the retrofitting of heat exchanger networks (HEN).
Use SPRINT software to explore how the characteristics of plate heat exchangers can be exploited for use in HEN retrofit.
As discussed, HEN can reduce utility requirements of process plants by recovering energy from appropriate process streams. Newly designed networks are optimised to maximise the heat recovery and minimise initial investment cost. Newly commissioned designs well understood and are less commonplace within industry. Historic assets were often designed before best practise knowledge on HEN synthesis was available and as a result are sub-optimal. This means the process is less efficient and energy is wasted. Wasting energy means the inefficient use of fuel. Money is spent on fuel that has its heat value wasted. In addition emissions released as a result of the process are greater. Industry has a responsibility with respect to the global warming and air pollution potential of their emissions. Increasing pressure from regulations restrict the quantities of industrial emissions.
Change in demands of the plant can mean the network is no longer appropriate for its requirements. A change in plant throughput can change process heat loads. This is often the case as industry expands to improve economic efficiency of assets. A change in the production specification will result in different process stream conditions. Changes in consumer demand often results in evolution and changes in product manufacturing. The effectiveness of a HEN would reflect on the production costs of the product.
Cost effective retrofitting methodologies are essential to improving the profitability of existing sub-optimal networks. The constraints inherent in the existing system make retrofitting often timely and expensive. Civil works require large amounts of space and production down time. The availability of space, capital investment and downtime constrains the repiping of exchangers, relocation of existing units, addition of new units, and addition of area to existing units. Gaining a greater understanding of the factors affecting the network retrofit is essential to producing a cost effective retrofit.
Retrofit can result in reduced utility requirements and reduced operating costs. However require an initial investment. Industry requires a practical approach that is easy to model, implement, and carry out.
Selection of appropriate heat exchanger types is often a decision left to engineering judgement as there is no current systematic procedure for exchanger type in retrofit design. In addition the use of plate heat exchangers in retrofit situations is rarely explored. There lacks a methodology for the use of plate heat exchangers to accompany current HEN retrofit design.
5. Literature Review
Existing literature on HEN retrofit is considered focussing on the key assumptions and achievements of major papers. Pinch as well as stochastic and deterministic mathematical methods are considered. Hybrid methods are also presented in which combinations of pinch and mathematical models are used to perform retrofit.
5.1. Pinch Retrofit Methods
Pinch retrofitting involves the adaptation of well-developed and understood pinch HEN design steps. This would involve targeting, an idea introduced by (Hohmann, 1971), in which the stream data is analysed to find optimum values for utilities and energy recovery characteristics. In pinch retrofit the designer is led through the process through the use of rules and constraints to aid and restrict the design. The end goal is the formulation of an optimal HEN.
The use of pinch approaches for HEN retrofit was introduced by Tjoe and Linnhoff (1986). The two-step method first uses the work by Townsend and Linnhoff (1984) on area efficiency and the area targeting formula, that was later progressed by Linnhoff and Ahmad (1990), to find the optimal minimum temperature difference (ΔTmin) for a HEN. The ΔTmin represents the minimum temperature difference required to provide the driving force for feasible heat transfer (Smith, 2005). The area targeting formula gives the minimum network area required at a specified ΔTmin. Finding the ratio of this area to the existing network area gives the area efficiency. This allows comparison of how efficiently the area can be used a different ΔTmin and allows the theoretical optimal ΔTmin to be selected. Constant area efficiency must be assumed across the network whereas in reality localised differences are inevitable. Decreasing ΔTmin results in greater heat recovery but greater area requirement, therefore ΔTmin is optimised to maximise heat recovery using the existing area.
During the second step the HEN is then re-designed. The pinch is identified as the point at which the ΔTmin between streams exists. Above and below the pinch areas are designed independently and the principle rules of pinch design are upheld. That is no hot utility below the pinch, no cold utility above the pinch, and no cross-pinch heat transfer. The existing structure is maintained but heat exchanger matches are re-arranged according to these rule. Although the method has the benefit of being easy to use, the area efficiency is assumed constant and the area distribution through the HEN is generalised. This means location within the network is not taken into account resulting in area target inaccuracies.
Shokoya (1992) aimed to improve area efficiency estimates by considering how the heat exchanger area is distributed through the network. For this an area matrix was formed. This considered the heat exchanger area between each hot and cold stream pair. Shokoya produced both an existing area matrix using known values and a target area matrix using information derived from hot and cold composite curves. Finding the difference between the two gives the deviation area matrix. The sum of the deviation area matrix values gives the target for the retrofit area. The method was able to improve on the work of Tjoe and Linnhoff (1986) and the resulting area targets were more realistic. Additionally the method could be used to aid retrofit with multiple utilities, one limitation of the former method. Despite this neither methods presented by Tjoe and Linnhoff (1986) nor Shokoya take into account the cost of topological modifications; this can result in overly complex designs being suggested as the optimal retrofit.
Avoiding the use of targeting, Carlsson et al. (1993) developed a cost based methodology for making design decisions during retrofitting. The approach was centred around the pinch considering the above and below pinch regions independently. A cost matrix is formulated for each design decision to consider the detailed implication of each choice. Piping distance, heat exchanger, auxiliary equipment, pressure drop, and maintenance costs are all included. The approach allows for comparison between different types of exchangers for each match. The consideration of the modifications to the overall retrofit cost was a major improvement on previous approaches. The work by Carlsson et al. remains one of the only papers to incorporate both heat exchanger type and location (van Reisen, 2008). Despite this important consideration for industrial applications, the large amount of data required to ensure accurate results makes the approach time consuming. The method is difficult to apply to large problems and requires an expert designer. Additionally as the process does not rely on targets, several different evaluations are required at different ΔTmin. When compared to other methods the resulting network is often different based on designer’s decisions.
Path analysis was proposed by van Reisen et al. (1995) using pre-screening to reduce the complexity of HEN retrofitting problems. Before retrofitting the HEN is broken down into subsections defined by potential energy savings. Each sub network is considered for retrofit looking at potential savings, complexity, practicality, and required investment. Sub-networks that offer attractive improvements for the required investment should be modified using either algorithms or design. The modifications are completed within the sub-networks before being considered in the network as a whole and comparing to retrofitting targets. The method proposed by Van Reisen et al. (1995) proved to greatly simplify complex retrofitting problem with the benefit of incorporating economic data. In some cases the method also resulted in a better retrofit than the work of Tjoe and Linnhoff (1986).
Further development of path analysis was published 3 years later by van Reisen et al. (1998). The progression used zoning to decompose the HEN and allowed consideration for structural interconnections. The methodology has the improvement of considering location, operability, and functionality. With the zones separated by structure, the benefit and cost of integrating different zones is considered. Decisions that offer attractive savings with appropriate initial investments are then chosen. Both the methods proposed by van Reisen and his colleagues have the benefit of highlighting the most beneficial modifications, this makes the approaches attractive to industry as it can be seen where the biggest improvements can be made.
Pinch analysis is shown to have limited use with retrofit design as it is much more suited to synthesis of new systems. In addition, for effective design it often requires an ‘expert’ user. Often the results given by pinch methods are overly complex as the models attempt to turn the existing HEN into a new grassroots design. For this reason the pinch methods can be useful for producing targets, or a best case design for a retrofit, however it is not always economic to achieve this design. Finally there are also often questions surrounding the use of minimum temperature approach and what value to use.
5.2. Mathematical Methods
HEN retrofit problems can also be tackled using mathematical models. The approaches discussed within this section perform the retrofit by translating the retrofit problem into a task to optimise. The optimisations are performed to minimise a given economic of modification criteria whilst meeting specified constraints. A range of optimising methods can be used, for that reason two sub-categories of mathematical methods have been formed. Deterministic methods comprise of non-linear programming (NLP), mixed integer linear programing (MILP), and mixed integer non-liner programming (MINLP). Stochastic approaches cover simulated annealing (SA) and genetic algorithm (GA) programmes. SA explores the optimising space through random selection of solutions before evolving towards a global optimum (Geltman, 2014). GA is modelled on observations from the natural world and represents natural selection and genetic mutation (Liu, et al., 2014). Superstructures are often used with mathematical approaches in which all possible retrofit designs are considered. It is then the job of the programme to find the design that best meets the objective whilst remaining within the constraints.
5.2.1. Deterministic Models
The first mathematical model targeted at solving HEN retrofitting problems was proposed by Yee and Grossmann (1986). The approach develops on the assignment-transhipment work of Papoulias and Grossmann (1983) and uses an MILP to find a retrofit solution with minimal structural changes whilst reaching a specified level of heat recovery. The approach has the benefit of not requiring large quantities of user input however the model is over simplified resulting in non-optimal solutions. The method uses the minimum new area and exchanger reassignments whilst maximising the use of existing heat exchangers. For this reason the suggested retrofit highly resembles the original HEN this has the benefit of reduced initial investment at the price of a sub-optimal solution. The approach formed is the basis of much further development.
The approach was further progressed by Yee and Grossmann (1991) and combines their previous work with the area targeting formula (Townsend & Linnhoff, 1984). The approach was capable of incorporating economic criteria and optimising for minimum annual cost. Two stages were used; pre-screening and optimisation. Comparable to targeting, the pre-screening step determines the optimal heat recovery for the HEN. Then the cost of utilities, topology changes, and additional area is evaluated against potential savings for each modification at different levels of heat recovery. For attractive modifications, they are put forward to the optimisation stage. This involves formation of a superstructure including all possible iterations of suggested new area. An MINLP then optimises to find minimum annual cost.
Ciric and Floudas presented a number of papers between 1988 and 1991 on deterministic approaches to HEN retrofit. Ciric and Floudas (1989) demonstrates and two stage approach first optimising the topology before the cost of the topology is then optimised. An MILP identifies the best modifications to the HEN structure in which consideration is given to all possible heat exchangers and matches. The second stage involves the formation of a superstructure that is then solved as an NLP problem. For this approach, additional area requirements and investment cost are rudimentary estimates. In addition, the desired heat recovery is fixed before the retrofitting and the approach must be iterated to find the best design. Testing this approach on examples given by the deterministic approach of Yee and Grossmann (1986) and the pinch approach of Tjoe and Linnhoff (1986) resulted in less new area requirements and consequently greater profits.
This approach was further developed to include structural modification costs and was combined into a single stage making use of an MINLP model (Ciric & Floudas, 1990a) (Ciric & Floudas, 1990b). Including improved area and cost calculations individual matches were considered in detail in contrast to the global area targets used in the former proposals. Despite the improvements the simultaneous approach of the “match-network hyper structure” resulted in complex programmes difficult to perform on computers of that time
Screening and optimising stages were adopted again in the cost based approach presented by Briones and Kokossis (1999). Two MILP models are implemented during screening. The first step, auditing, considers additional matches and adding area to existing matches. Costs are estimated and assigned to potential modifications. The second stage, unit development, involves the consideration of existing HEN structural modification; relocation and repiping costs. Hypertargets are formed with the results from the screening stage. These act similar to targets but are more complex. They have the benefit of not relying on area efficiency and can include process constraints unlike targets for pinch methods. The optimisation uses the hypertargets to optimise the topology for relevant objectives such as minimum, additional area, annual cost, or investment cost. The optimisation is performed by an NLP model and the overall approach has the benefit of including capital, operating, and structural modification costs.
A two-step methodology proposed by Ma et al. (2000) uses a combination of linear and non-linear models in order to find the global optimum for HEN retrofitting problems. Step one involves the assumption of a Constant Approach Temperature (CAT) for all HEN matches. This is justified on the premise that the temperature driving force must be shared equally between matches. This allows linearization of the heat transfer area calculations and means the solution can easily be found with an MILP model. The linearity of the model means it cannot be trapped by a local optimum as can be the case in non-linear models shown in Figure 3. The CAT model adopts same superstructure that was used in the approach given by Yee and Grossmann (1991). The linear model does not always give a feasible solution but does move close to the global optimum. For this reason the structure proposed in step 1 acts as a good starting point for the MINLP model deployed in step 2. The nonlinear model simultaneously handles energy recovery, structural modifications, and heat transfer area. The model also gives additional consideration to exchanger area that was estimated in step 1.
Sorsak and Kranvanja (2004) proposed a simultaneous MINLP approach capable of incorporating different types of heat exchangers. The method is a development of their 2002 approach to HEN synthesis and is an extension of the superstructure work of Yee and Grossmann. The inclusion of different heat exchangers brought challenges with regards to heat transfer and exchanger geometry and results showed that feasibility of networks was highly influenced by exchanger type. Plate and frame, two pass shell and tube, and double pipe heat exchangers were included within the model. Although requiring greater computational time, the method using multiple exchangers produced a more optimal HEN retrofit when compared to a single type retrofit in the study (Sorsak & Kravanja, 2004). This approach also has relevance to this study as it introduces the first use of plate heat exchangers to retrofitting models.
Ponce-Ortega et al. published two papers introducing the relationship between process conditions and heat integration (2008a) and introducing phase change (2008b). Both use MINLP and aim to improve on limitations of previous approaches. The first considers changing process conditions and heat integration simultaneously (Ponce-Ortega, et al., 2008a); where previous models have fixed operating conditions. It is no surprise that this new consideration improved the potential for heat recovery resulting in more optimal retrofit solutions. The second approach considers streams with both sensible and latent heat changes (Ponce-Ortega, et al., 2008b). The consideration of phase change makes the approach more appropriate for use within industry. The first paper proposes to maximise retrofit profit (Ponce-Ortega, et al., 2008a) where the second aims to minimise annual cost.
Nguyen et al. (2010) proposed a one-step MILP model that allows for user decision making on stream splitting, number of new exchanger units, and exchanger relocation. The approach develops the MILP model for grassroots HEN synthesis proposed by Barbaro & Bagajewicz (2005). The appraoach uses trans-shipment, in which the hot and cold streams are segmented and the individual segments are considered for heat exchange from hot to cold. The optimisation simultaneously the heat exchange area between suggested pairs and the network structure (Nguyen, et al., 2010).
A two stage optimisation is proposed by Pan et al. (2013) in a typical MINLP is converted to an MILP based iterative method. Using the concept of pinching match is used to more efficiently produce the network superstructure used for retrofit design. The resulting superstructure is smaller and requires less time to generate. In addition the problems associated with non-linear programming are negated owing to the linear based iterative method. After the formation of the superstructure there are two optimisation stages. The first optimises the structure with the latter optimising the investment required. The approach proves to be efficient and perform better retrofit designs then previous methods and has the benefit of being appropriate for larger, industry sized problems.
Deterministic approaches are able to provide retrofit design with little user interaction however still provide a level of engineering decision making and control. Non-linear models have shown difficulties with obtaining globally optimal designs whilst linear models can sometimes produce overly complex or unfeasible designs. Automation of retrofitting problems has the huge advantage of reducing retrofitting time however the ‘black box’ nature of the programmes gives less insight to the solutions presented.
5.2.2. Stochastic Models
Nielsen et al. (1996) used a simulated annealing (SA) model to perform a complex retrofitting procedure. Their methodology would allow modelling of a range of heat exchangers, variable heat transfer coefficients, pressure drop, and a range of additional equipment. The object orientated model that resulted was very flexible but on testing it was found hard to find optimal solutions. For this reason it was suggested by the authors to use in combination with a direct search algorithm. In the approach proposed by Athier et al. (1998) two levels are used combining a SA model and an NLP formulation. In the first level the topological structure is optimised. This is done iteratively by the SA model. In the lower level the NLP optimises the area that is required in new units and additional area for existing units. The model applied is simpler than the previous work of Nielsen et al. (1996) for that reason the model was appropriate for reasonably sized systems and their results were found to be robust.
Bochenek & Jezowski (2006) used an evolutionary genetic algorithm (GA) with two-levels to find optimal solutions to retrofitting problems. A structural matrix is used to represent all of the structural features of the heat exchanger network. This allows GA to optimise as a single problem. The first level optimises the network structure where GA optimises the heat exchanger topology and locations of stream splitting. Then, split ratio and areas of the heat exchangers is found in the second level. The method was computationally demanding despite simplifying the network into a matrix and required long computational times. Alone, GA was not effective at dealing with continuous variables. To improve the GA approach Razaei & Shafiei (2009) sought to combine with deterministic NLP and integer linear programming (ILP). Continuous variables previously handled poorly were optimised by the NLP formulation. Optimal split ratios and minimum approach temperatures are found using an easily computed search loop. The ILP aims to find the optimum cost required for the alterations and handles the introduction of new units or reuses of existing ones.
Zhang and Rangaiah (2013) proposed a one-step approach centred on a stochastic model that dealt with continuous and discrete variables in combination. Being trapped by a local optimal as discussed in deterministic methods, section 5.2.1, is avoided by simultaneously optimising the network structure and system parameters; this results in a global optimum. An integrated differential evolution model (IDE) is implemented and was shown to greatly improve computational efficiency when compared to previous two-step methods.
Hybrid GA is used in an approach proposed by Liu et al. (2014) with the goal of reducing the computation capacity required to solve HEN retrofit problems. The existing structure and heat exchangers are fully employed minimising topological modifications. The HEN is optimised to produce minimum cost of structural modifications and to minimise utility cost.
The concept of optimising HEN retrofit with variable heat capacity is developed by Sreelatha & Rangaiah (2015). Previous approaches use step-wise models to replicate changing heat capacity however a polynomial fitting formulation is used to reproduce the gradual changes seen in reality (Sreepathi & Rangaiah, 2015). Both multi-objective and single objective optimiser models are used. This gave the approach a great deal of flexibility allowing control over new area to existing exchangers, new units, and structural modifications. In additional through use of the multi-objective optimisers, many solutions were given allowing user interaction to select depending on circumstance. All suggested solutions were pareto-optimal; meaning an increase in one objective would detract from another. The approach
SA has the advantage of not getting caught at local optimums by using enough random selection early on. However unlikely to find completely optimal solution will produce a good solution (wolfram)
5.3. Hybrid Methods
Hybrid methods aim to form a synergy between pinch analysis and mathematical programming in order to perform HEN retrofitting. They combine the user interaction of pinch analysis with the automated nature of the mathematical programming to reduce retrofit time and improve results. Separation of the design problem can be achieved using aspects of the pinch analysis before optimisation and decision making aided by mathematical programming. Hybrid methods also have the advantage of being able to handle some of the larger industrial problems often not appropriate for non-linear optimisation.
Pinch often assumed area efficiency of existing and new area is the same – we know this to be a limiting assumption. In addition pinch methods often require expert users and require large amounts of resources and time.
Work produced by Asante and Zhu (1996)(1997) proposed and developed on combining user interaction and mathematical programming. The two stage process used diagnosis and optimisation to highlight HEN bottlenecks before proposing appropriate solutions to these limitations.
During diagnosis the topology is fixed and the heat exchanger duties and stream temperatures are optimised. As the energy recovery increases with optimisation, pinches within the system will be highlighted as heat exchangers with at least one temperature difference as zero. These pinches represent bottlenecks that are single points limiting the entire performance. The bottlenecks must be overcome to increase energy recovery. Asante and Zhu (1997) suggest an MILP formulation that will suggest the best modification to overcome specific bottlenecks. Stage two then combines the most attractive modifications and performs an optimisation using an NLP to minimise the retrofit cost. The approach aims to minimise structural changes by evolving from the existing structure and identifying the most important changes to reach the energy target.
6. Plate Heat Exchangers
Plate heat exchangers use flat sheets of metal between heat transfer fluids to achieve transfer of energy from one to the other. The hot and cold fluids pass through the alternative spaces between plates as can be seen in Figure 1. The flow of the hot and cold fluid is truly counter-current (1). Plate heat exchangers differ from the more commonly utilised shell and tube type in a number of ways; from their features and design methods to how they are implemented. These differences can be exploited to best serve a design requirement. Plate heat exchangers will be further explored and their use in retrofit design discussed.
Relative to shell and tube heat exchangers, plate heat exchangers have a greater heat transfer coefficient, U (1). This is because the thin plates offer little resistance to heat transfer, the process fluid velocity is maintained high, and the flow pattern turbulent (2). In addition the flow of process fluids is counter current; this is the most efficient for heat transfer as the log mean temperature difference between fluids is at its greatest (3). For this reason compared to shell and tube heat exchangers, plate heat exchangers are far more efficient. Therefore less area is required for the same transfer of energy.
The physically compact nature of plate heat exchangers brings with it many benefits. The small unit footprint lends itself well to plant retrofit; plate exchangers can offer an 80% reduction on footprint for the same transfer area to shell and tube (1). Often space is limited in existing assets, and it is hard to physically fit a shell and tube exchanger in. As plate heat exchangers can fit a large amount of exchange area within a small space and because they are much more efficient they can sometimes be used where a shell and tube could not.
Plate heat exchangers can be plate and frame where they are bolted together to allow easy disassembly. Whereas welded or brazed plate heat exchangers are fixed and cannot be taken apart for thorough cleaning (4). Cleaning is therefore left down to chemical and high pressure cleaning in place. For this reason they may not be appropriate for industrial applications in food and pharmaceuticals or where liquids are highly fouling (4). However, due to their greater structural integrity, welded flat plate heat exchangers can handle a greater range of operating conditions. Higher pressures and temperatures are appropriate where they would not be for plate and frame exchangers that rely on gaskets to create seals between adjacent plates. Despite this, plate and frame heat exchangers are popular as a result of their flexibility (1). Additional area can easily be added by the introduction of more plates. Also, because they can be taken apart, they can be thoroughly cleaned or maintained if necessary.
Despite not being able to deal with extremely high temperatures, plate heat exchangers can perform well even with small temperature differences between hot and cold fluids. This makes them attractive with respect to retrofit as it allows stream matches where shell and tube exchangers would not be effective. Plate and frame are low cost (1) in comparison to shell and tube heat exchangers; this again lends itself well to retrofit situations. A smaller capital investment makes it more attractive to an existing HEN attempting to increase energy recovery. One limitation of plate heat exchangers that limits its use within HEN retrofit is the high pressure drop (3). As a result of the flow pattern within the exchangers they can high very high pressure drop. Existing HEN can often be greatly constrained by pressure drop as this will affect downstream processes. Additional auxiliary equipment and pumping operational costs can be expensive additions to plant total annual cost. Finally, turbulent and high velocity flow patterns of the process fluids mean plate heat exchangers are good at dealing with fouling fluids. But, as discussed, welded type exchangers cannot be opened and cleaned in the case of extreme fouling. Exchangers can also be specifically designed for phase change, highly corrosive fluids, highly viscous fluids, and fibrous or particulate fluids (1).
Plate heat exchangers offer engineers a huge scope of design options. Plate design has attracted a huge amount of attention and affects the flow of the process fluids. This affects the velocity and turbulence of the flow therefore affecting the Reynolds number and consequently the film side heat transfer coefficient. Commonly used are chevron designs. Stainless steel is often used as it is corrosion resistant, high heat transfer while maintaining its temperature resistance.
Assuming flow is separated equally between plate channels
For design of heat exchangers the heat load is first found, as shown by Equation 1. This represents the heat to be transferred. It can be found by taking the mass flow rate, m, heat capacity, CP, and temperature difference of one of the streams,
δT. Overall heat transfer coefficient can be estimated from literature depending on the process fluids. Then rearranging Equation 2 inputting values for log mean temperature difference, ΔTLMTD, and temperature correction factor, FT, the area, A, can be found. The correction factor acts to adjust for non-idealised flow. In the case of shell and tube this has a large bearing on the area calculation as flow is partially counter current. With plate heat exchangers the flow is almost ideal counter current and values of FT are as high as 0.98.
|Q=mCPδṪ||Equation 1 – Heat Load|
|Q=UOAΔTLMTDFT||Equation 2 – Heat Transfer|
|ΔTLMTD=ΔT1-ΔT2lnΔT1ΔT2||Equation 3 – Log Mean Temperature Difference|
ΔTLMTD represents the effective driving force of heat transfer within the exchanger and is calculated by Equation 3. The area calculated represents the total heat transfer area. The find the number of plates this must be divided by the plate surface area. This is a design choice and is affected by flow rate, temperatures, pressures and availability of the plates.
The design process can by much more detailed where the velocities of the respective hot and cold fluids are found using plate spacing’s to find cross sectional area then the fluid velocities. This then allows calculation of the Reynold, Ru, and Nusselt, Nu, numbers allowing calculation of the respective film heat transfer coefficients. From this the overall heat transfer coefficient can be calculated taking into account fouling factors and thermal conductivity of the plate medium. By then using this newly calculated heat transfer coefficient it is compared to the initially assumed value. If the values are far from the initial guess the number of plates should be adjusted until the calculated value is within an acceptable range of the estimated value.
7. Work Plan
Gantt chart w/ annotation
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