Ancillary Piping Equipment

Unit Objective

There are six Units in Module 4. Unit 1 focuses on Introduction to Pipe Installation and Safety, Unit 2; Piping Services, Unit 3; Electricity on Site, Unit 4; Bracket Fabrication, Unit 5; Ancillary Piping Equipment and Unit 6; Piping system assembly.

In this unit you will be introduced to good practice guidelines for installing ancillary piping equipment such as pumps, heat exchangers and valves and how best to orientate and bracket piping coming to and from this equipment.

Learning Outcome

By the end of this unit each apprentice will be able to:

·         Identify and describe the main ancillary piping system components

·         Identify and select the correct pump for the three most common pumping applications.

·         Explain why pipe lines are installed at low level, close to walls and accessible to read instruments wherever possible.

·         Explain why safe access to equipment (e.g. heat exchangers and pumps) is important during commissioning, maintenance and servicing.

·         List reasons why valves are used in piping systems.

·         Outline the importance of bracketing pipe work around equipment to facilitate safe removal of equipment for maintenance and to ensure that piping does not strain the equipment.

·         Describe the standard procedure for safe start-up and commissioning of ancillary piping equipment.

·         Recognise the importance of and the need to retain and file equipment manuals and material certification.

1.0   Ancillary Piping Components

1.1    Identify Ancillary Piping Components

Ancillary piping components are the additional items installed in a piping system such as pumps, heat exchangers, valves and instrumentation.  Their requirements vary depending on the media being transported in the piping system.  Module 3 unit 2 has dealt with pumps, valves and basic instrumentation so this module will examine the different type of common heat exchangers and evaluate different pumps against specific pump selection criteria.

1.2Types of Heat Exchangers

A heat exchanger is a device built for efficient heat transfer from one medium to another. The heating or cooling media is separated from the product to be heated or cooled by a solid wall, so that they never mix.  There are three primary classifications of heat exchangers according to their flow arrangement. In parallel-flow heat exchangers, the two fluids enter the exchanger at the same end, and travel in parallel to one another to the other side. In counter-flow heat exchangers the fluids enter the exchanger from opposite ends. In a cross-flow heat exchanger, the fluids travel roughly perpendicular to one another through the exchanger.  The counter current design is most efficient, in that it can transfer the most heat from the heat (transfer) medium.  For efficiency, heat exchangers are designed to maximize the surface area of the wall between the two fluids, while minimizing resistance to fluid flow through the exchanger. The exchanger's performance can also be affected by the addition of fins or corrugations in one or both directions, which increase surface area and may channel fluid flow or induce turbulence.  We will deal the 2 most common types of heat exchangers:

·         Shell and Tube Heat Exchanger

·         Plate Heat Exchanger

1.3 Shell and Tube Heat Exchanger

Shell and tube heat exchangers consist of a series of tubes (see Figure 1). One set of these tubes contains the fluid that must be either heated or cooled. The second fluid runs over the tubes that are being heated or cooled so that it can either provide the heat or absorb the heat required. A set of tubes is called the tube bundle and can be made up of several types of tubes: plain, longitudinally finned, etc. Shell and Tube heat exchangers are typically used for high pressure applications (with pressures greater than 30 bar and temperatures greater than 260°C).  This is because the shell and tube heat exchangers are robust due to their shape.

There are several thermal design features that are to be taken into account when designing the tubes in the shell and tube heat exchangers. These include:

·         Tube diameter: Using a small tube diameter makes the heat exchanger both economical and compact. However, it is more likely for the heat exchanger to foul up faster and the small size makes mechanical cleaning of the fouling difficult.

·         Tube thickness: The thickness of the wall of the tubes needs to be considered for the following factors; flow rates, pressure ratings and corrosion requirements.

·         Tube length: heat exchangers are usually cheaper when they have a smaller shell diameter and a long tube length which reduce the labour for manufacture.  However this must be considered in conjunction with space on site to install and the need to withdraw the tube bundle for servicing.

·         Tube pitch: (i.e., the centre-centre distance of adjoining tubes) needs to be considered as a large tube pitch leads to a larger overall shell diameter which leads to a more expensive heat exchanger while a narrower tube pitch can cause inefficient heat transfer.

·         Tube corrugation: this is mainly used on the inner tubes to increase flow turbulence and heat transfer giving a better heat exchanger performance.

·         Baffle Design: baffles are used in shell and tube heat exchangers to direct the fluid in the shell across the tube bundle. They run perpendicularly to the shell and hold the bundle, preventing the tubes

from sagging over a long length. They can also prevent the tubes from vibrating.  (See Figure 2)

Figure 2 – Tube bundle with baffle plates

1.4 Plate and Frame Heat Exchanger

A plate heat exchanger is composed of multiple, thin, slightly-separated plates that have very large surface areas and fluid flow passages in between for heat transfer.  

Figure 3 – Plate and frame heat exchanger with media connections on head plate

Plate heat exchangers can differ in the types of plates that are used, and in the configurations of those plates. Some plates may be formed with "chevron" (Figure 4a) or other patterns to increase flow turbulence and therefore heat transfer, where others may have machined fins and/or grooves.  The gasket design (Figure 4a) allows the heating or cooling medium to flow through every second space in the plate stack and the product medium to be heated or cooled flows through the alternate spaces as can be seen in Figure 4b.

Figure 4a – Chevron plate to improve heat transfer, gasket design to direct flow

 Figure 4b - Counter current flow in alternate fluid spaces

While shell and tube heat exchangers are more suited to high pressure applications plate and frame heat exchangers have the following advantages:

·         Reduced installation footprint, weighs less and delivers higher performance

·         Efficient operation with up to 98% heat recovery or regeneration reduces energy costs

·         Low liquid hold-up enables faster reaction times to change in process

·         Fluids flow counter-current to each other between the parallel passages in each pass

·         Full access to both sides of the heat-transfer surface for inspection, maintenance, and cleaning

·         Access is readily accomplished within the installed space of the unit, therefore there is no need to allow for additional “withdrawal” room.

·         Modular design enables expansion of your heat exchanger as process requirements grow

1.5 Pump Selection Criteria

Pumps transfer liquids from one point to another by converting mechanical energy into pressure energy (head).  The pressure applied to the liquid forces the fluid to flow at the required rate and to overcome friction (or head) losses in piping, valves, fittings, and process equipment.   Pumping applications include constant or variable flow rate requirements, serving single or networked loads, and consisting of open loops (liquid delivery) or closed loops (recirculation systems).  When selecting a pump the following points should be considered:

The pumping system designer must consider fluid properties, determine end use requirements, and understand environmental conditions.

·         Pumping rate or flow rate required by the system, factors to be considered are the usage profiles of the users and the storage capacity built into the system.

·         Minimum available net positive suction head (this requires knowledge of the maximum lift required and all head losses on the intake side of the pump).

·         The discharge pressure required at the point of use, on top of this flow characteristics of the liquid, friction losses in the system and any head heights that the pump must overcome need to be considered.

·         Characteristics of the fluid to be pumped (e.g. viscosity, temperature, solids content, corrosiveness, etc.).

·         Availability of suitable power to drive the pump.  In some instances in a solvent laden ATEX area, a pneumatic diaphragm pump is used as there is no electrical requirement to power the pump.

·         Pump location, (e.g., indoors, outdoors, submerged, in a corrosive environment)

·         Servicing / maintenance requirements of the pump and availability of spares.  This will also be affected by the operating conditions of the pump and if it is operating 24/7.

Once these and perhaps other site-specific factors are known, it is possible to consult manufacturers’ literature and consider the available pumps. A major portion of this process involves consideration of trade-offs among the reliability, first cost, and operation and maintenance cost of various pumps having suitable flow/head/efficiency characteristics.  Table 1 below highlights the advantages relative to each other of the following 3 pumps and why they would be selected for a particular duty:

·         Centrifugal pumps

·         Diaphragm pumps

·         Drum pumps

Pump Characteristic	Centrifugal	Diaphragm	Drum
Flow Rate	High	Medium	Low
NPSH	Needs a positive head	Can suck liquid into pump from below	Can suck liquid into pump from below
Pressure	Normally low	Medium	High
Viscosity/ solids content	Low	High	Medium
Power supply	Electrical	Compressed Air	Electrical or Compressed air
Location	Supplier dependant	Supplier dependant	Supplier dependant
Maintenance	Low	Medium	Medium

Table 1 – Comparison of pump characteristics for pump selection

2.0  Piping Installation

2.1 Design of Equipment Layout/Pipe Routing

First the basis of design is established, the equipment and materials of construction selected and the Process Flow Diagram (PFD) agreed for a process piping system.  The next step is to move on to a more detailed design of the system.  The P&ID provides a schematic layout of the equipment, valves, instrumentation and line sizes.  It is however not drawn to scale and only present the relationship or sequence between components and how they interact to control the systems functions.

The physical layout of the major items of equipment, valves and instrumentation is vital to the ergonomic operation of the plant long after the construction phase is complete.   The interconnection pipework and bracketing of same is also critical to facilitate ease of access and future maintenance of the system.   While this list is not exhaustive the following points should be considered when finalizing equipment and piping layouts:

·         Adequate space for personnel to access system monitoring instrumentation and to access equipment to regularly inspect for signs of leaks or wear.

·         Space for removal of internal components for maintenance (e.g. tube bundles from heat exchangers or agitators from the top of a vessel.)

·         Thermal expansion and contraction of short runs of pipework in a plant room can in many instances be catered for by avoiding routing pipe in straight lines and introduce bends instead.

·         Pipe routing should utilize the surrounding structure for support where possible. Horizontal and parallel pipe runs at different elevations should be spaced for branch connections and also for independent pipe supports.

·         Sufficient and well designed bracketing should ensure that no undue stresses or forces are transmitted to system equipment which may cause premature failure of bearings etc.  Supports are also important to retain pipework and facilitate ease of removal of equipment for servicing.

·         Consideration should be given to other trades when installing pipe and should be coordinated to accommodate electrical conduit requirements, civil requirements for drains and clearances for pipe insulation where required.

The layout of equipment and pipe routing is greatly aided by the use of 3D modeling software (see Figure 5a and 5b) which allows piping designers to visualize the complete installation and zoom in on congested areas to check for clashes of valves and instrumentation.  Extensive component libraries allow the designer to quickly import standard components to compile the system model which can then perform static and dynamic stress analyses on pipe and equipment and indicate the best positions for anchors and supports.   Individual isometrics can be exported with bills of materials for purchasing to procure the necessary materials and the more sophisticated packages can be linked with ERP systems to provide a complete costing tool.

2.2 Valve Selection

For liquid piping systems, valves are the controlling element. Valves are used to isolate equipment and piping systems, regulate flow, prevent backflow, and regulate and relieve pressure. The most suitable valve must be carefully selected for the piping system. The minimum design or selection parameters for the valve most suitable for an application are the following: size, material of construction, pressure and temperature ratings, and end connections. In addition, if the valve is to be used for control purposes, additional parameters must be defined.  These parameters include: method of operation, maximum and minimum flow capacity requirement, pressure drop during normal flowing conditions, pressure drop at shutoff, and maximum and minimum inlet pressure at the valve. These parameters are met by selecting body styles, material of construction, seats, packing, end connections, operators and supports.

2.3Ergonomics of Piping Design

While construction schedule and costs drive mechanical contractors to install piping systems quickly it must be remembered that the final system will be operated for many years in the future.  For this reason it is critical that proper consideration be given to the ergonomic layout of the equipment and instrumentation so as to ensure that the operator can comfortably operate the system.  Simple things such as gauges at an ergonomic height, orientated in an upright position and with readable sized scales can make the monitoring and recording of information so much easier.  Ease of access to equipment for periodic inspections and checking for leaks will ensure that the plant is well cared for and well maintained.  Space for removing equipment

`2.4  Case Study of a Centrifugal Pump Set Installation

Figure 6 above shows the components utilized on a typical centrifugal pump installation and the following gives a brief outline on why they are required for an effective centrifugal pump set.

Isolation Valves

Pumps often need repairs. Sometimes mechanical seal leakages occur. Pump bearings also need replacement. In order to carry out such repairs pump needs to isolated.  It should have no process fluid so that it can be worked upon.  Installing isolation valves ensures that no liquid flows towards the pump due to gravity from either upstream or downstream of pump.

Strainer

A strainer is a 'filter' that prevents undesirable solid particles to flow upstream and clog the equipments. A strainer contains a mesh that prevents the particles from flowing through it. Two main types of strainers are: y-type strainers and basket strainers. Y-type strainers (shown in figure 6) are used for relatively clean fluids while basket type strainers are used where greater amounts of particles are present.  When running a pump for the first time it is vital that the strainer is checked on a regular basis as it will often clog up with construction dirt and debris left over from the system fabrication.

Check valve

Check valves are used to prevent backflow in the system, if a pump was to malfunction the fluid which has been pumped upstream would try to flow back towards the pump. In order to stop this from happening a check valve is used.

3.0  Piping Systems Commissioning

3.1 Process Commissioning

Process commissioning occurs between the time construction is complete and plant startup commences.  During this period process commissioning personnel are occupied with the task of ensuring the facilities have been constructed and assembled according to the engineering design and the equipment manufacturer’s directions.  The objective is to ensure that the equipment has been properly installed and is ready to receive process materials and operate as originally conceived.

While this list is not exhaustive the following points should be considered when preparing for safe start-up and commissioning of ancillary piping equipment:

·         Commissioning plan and procedures must be prepared that describe in detail how the various tests will be conducted and evaluated.

·         A commissioning team assembled of experienced managers, engineers, plant operators and fitters (these may be from the mechanical contractor who installed the facility) acting as support staff.

·         The process commissioning procedures must also describe safety precautions that must be taken before, during, and after the commissioning process.

·         The detailed commissioning plan should synchronize the turnover of process units from the mechanical contractor to the commissioning team.

·         Before the commissioning team will accept any packages the following should be signed off and available :

a.       DQ and basis of design for each of the systems in the facility

b.      P&ID System walk-downs

c.       Flushing and pressure test, test packs.

d.      Chemical cleaning and passivation

e.       IQ and OQ documentation

f.        Equipment suppliers documentation and Operation and maintenance (O&M) manuals

·         Project planning software, should be used to organize and control the commissioning process, so that resources can be scheduled and deployed in the most effective way to ensure that all elements are eventually commissioned and declared operable.

·         It is essential that there is accurate progress reporting and feed back from the field, as more often than not the commissioning of the next system is dependant on the first being a success.

Just as with the actual construction of the process facility, process commissioning is a complex activity covering all aspects of the newly constructed facilities.

·         Each element of the process unit is examined and tested.

·         Process control valves must be stroked,

·         Controller tuning coefficients must be checked.

·         Sensors and analyzers must be calibrated

·         Relief valve settings must be checked,

·         Piping and equipment is often hydrostatically tested or tested with inert gas again to find and eliminate any leaks which may occur from final assembly or fitting of sensitive instruments which were removed for the system pressure test.

·         Strainers, filters and tramp metal collectors must be installed at critical locations in the piping system to prevent damage to pumps and control valves.

Coordination between trades is essential and tasks such as the following will need two or more trades to verify:

·         Insulation must be inspected and steam tracing tested.

·         Electrical connections must be checked and electrical equipment tested where and when it can be done safely.

·         Rotating equipment must be checked for alignment and manually rotated to ensure there are no interferences. Electrical motors need to be run to ensure the connections are correctly installed and that the motor rotates in the correct direction.

Should problems develop during the startup phase, written plans and procedures should be in place to empty each process unit in a safe and environmentally compliant manner so that whatever problems occurred can be fixed.  Any defects found during he commissioning process must be corrected by the contractor before the process unit commissioning can be declared as complete.

A major part of process commissioning is in preparing the operating instructions for the startup of the process. The procedures for a newly constructed plant often differ from the procedures that would be placed in service after a successful production campaign. In the case of a newly constructed plant, the procedure may call for each upstream unit to be brought up to operating temperatures and pressure and held for a period of time to validate the integrity of the unit before process material is allowed to flow to the next downstream unit.

3.2 Piping systems Documentation and Validation

More and more industries have placed an increasing emphasis on quality standards and documentation in order to expedite their approval process for either their own internal corporate quality requirements or for external bodies such as the IMB (Irish Medicines Board) or the FDA (United States Food and Drug Administration).  The approval process requires that the facility in which a new product or drug is produced must be designed, constructed and commissioned so that it meets the criteria for process validation.

Validation is the action of proving, in accordance with the principles of GMP (Good Manufacturing Practice), that any procedure, process, equipment, material activity or system consistently leads to the expected results. Documented evidence provides a high degree of assurance that a specific system, equipment or process will consistently produce a product meeting its pre-determined specifications and quality attributes. To put it simply, validation is nothing more than proving that a process actually works. 

Failure to achieve validation on the first attempt can be very costly to the facility owner, so maintaining quality from the design phase throughout the construction process is essential.  To this end the pipe fitter / welder can play a major part in ensuring the following documents are maintained and collated in a controlled fashion:

·         Collecting and filing material certification for goods received on site

·         Collecting and filing equipment documentation and manuals for equipment received on site

·         Use the correct tacking and weld procedures and ensure that all personnel welder qualifications are kept current.

·         Accurate maintenance of weld record sheets and isometrics during the fabrication and installation phase.

·         Co-ordinate with inspection companies to ensure weld inspection is maintained at or above the required % level or that all welds are examined where 100% traceability is required.

·          Ensure test packs are completed in the required format and signed off and witnessed by the relevant personnel

·         Ensure any re-routings or changes in drawings are recorded properly and that the information is relayed through the proper channels to ensure accurate “As-built” drawings are handed over to the client.

Exercises

·         Identify 2 advantages and 2 disadvantages of a plate and frame heat exchanger

·         Identify 2 reasons as to why a centrifugal pump would be chosen before an air operated diaphragm pump

·         List 3 reasons why good equipment and piping layout planning is vital for the operation of a facility after handover.

·         Dismantle and examine a pipe strainer and explain how it protects equipment upstream.

·         Identify 3 ways how a pipe fitter can contribute to the documentation process for system validation.

Mechanical Engineering 370 Thermodynamics

1    A classroom that normally contains 40 people is to be air conditioned with window air conditioning units of 6 kW cooling capacity.  A person at rest may be assumed to dissipate heat at a rate of about 360 kJ/h.  There are 10 light bulbs in the room, each with a rating of 100 W.  The rate of heat transfer to the classroom through the walls is 15,000 kJ/h.  If the room air is to be maintained at a constant temperature of 21^oC, determine the number of air-conditioning units required.

In this problem, we define the system to be the air in the room.  We can assume that the air behaves as an ideal gas.  If the temperature remains constant, there is no change in the internal energy of the air.  Since the volume of the room remains constant, there is no work done.[1]  Thus the first law reduces to the equation that Q = 0.  We account for the different heat sources in the total heat rate, Q, as follows.  The heat input comes (a) from the 40 students, Q_s = (40 people)(360 kJ/h•person)(kW•s /kJ)(h/3600 s) = 4 kW, (b) heat transfer through the wall, Q_w = (15,000 kJ/h)(kW/kJ•s)(h/3600 s) =4.167 kW, and (c) heat added by the light bulbs, Q_b = 10(100 W) = 1000 W = 1 kW.  Thus the total heat input is 4 + 4.167 + 1 = 9.167 kW.  The heat removal comes from the air conditioners which remove 5 kW each.  Thus for the net Q to be zero we must have N air conditioners such that (5 kW)N = 9.167 kW.  To satisfy this requirement, we must have two air conditioning units.

2    A 0.5 m³ rigid tank contains regrigerant-134a initially at 160 kPa and 40% quality.  Heat is now transferred to the refrigerant until the pressure reaches 700 kPa.  Determine (a) the mass of refrigerant in the tank and (b) the amount of heat transferred.  Also, show the process on a P-v diagram with respect to the saturation lines.

The mass of refrigerant may be found from the initial state using the equation that m = V / v₁ where V = 0.5 m³ and v is found from the temperature and the quality.  Since we are given an initial quality, x₁ = 40%, we know that we are in the mixed region.  The specific volume is found from the specific volumes of the saturated liquid and vapor, which are found in Table A-12 on page 930: v_f(160 kPa) = 0.0007437 m³/kg and v_g(160 kPa) = 0.12348 m³/kg.  We then find the initial specific volume as follows.

With this specific volume, we then find the refrigerant mass as follows:

[1] Note that is both the mass and volume remain constant, the specific volume remains constant.  Since both the temperature and specific volume are constant, the internal energy is a constant, regardless of whether or not we assume that air is an ideal gas.

To compute the heat transfer we apply the first law, Q = DU + W.  We assume that there is no volume change in the “rigid” tank.  If there is no volume change, no work is done.  With W = 0, Q = DU.  We find DU = m(u₂ – u₁) where the specific internal energies are found from the property tables.  At the initial state we find u from the quality in the same way that we found the volume.

Because this is a constant volume process, the final state has the same specific volume as the initial state (0.049838 m³/kg) and a given pressure of 700 kPa.  Knowing these data we can plot the constant volume path for this process as shown in the figure at the left.  We see that the initial state (1) is in the mixed region and the increase in pressure, at constant volume, brings the final state into the gas region.                                        Plot for problem 2

When we try to find the final state (P = 700 kPa, v = 0.049838 m³/kg) in the superheat table, A-13, on page 930, we see that the final specific volume in the table for P = 700 kPa = 0.70 MPa (at a temperature of 160^oC) is only 0.048597 m³/kg.  Furthermore, the specific volume is increasing with temperature so that the final state is at a higher temperature than the final temperature in the table.

The internal energy at the final state can be found by extrapolation beyond the last value in the table, using the same formula as for interpolation, with the last two points in the table.

We can now find the heat transfer as follows.

3    A 20 ft³ rigid tank contains saturated regrigerant-134a vapor at 160 psia.  As a result of heat transfer from the refrigerant, the pressure drops to 50 psia.  Show the process on a P-v diagram with respect to the saturation lines, and determine (a) the final temperature, (b) the amount of refrigerant that has condensed, and (c) the heat transfer.

The solution of this problem is similar to the previous one.  It is a constant volume process so that the work is zero and the heat transfer equals the change in internal energy.  The P-v diagram for this problem is shown below.  The process starts at the saturated vapor line at 120 psia and as heat is removed, some refrigerant condenses, moving the process into the mixed region at the final state where P₂ = 50 psia..

Since the final state is in the mixed region, we know that the final temperature is the saturation temperature at the final pressure of 50 psia.  From table A-12E on page 977, we find that the final temperature, T_sat(50 psia) = 40.23^oF.

3

The amount of refrigerant that has condensed is the total mass minus the mass that is vapor at the final state.  This is m – x₂m.  We can find the total mass from the specific volume at the initial state and the total volume of the container: m = V/v₁, where v₁ is the volume of the saturated vapor, v_g, at the initial pressure of 160 psia.  From Table A-12E on page 977 we find that v_g(160 psia) = 0.29316 ft³/lb_m, so that the total mass is given by the following equation.

                                                                      Problem 3

The specific volume at the final state is the same as the initial specific volume, so we can find the quality at this final state from the saturated liquid and vapor specific volumes at the final pressure of 50 psia, taken from Table A-12E.

The mass condensed can now be found.

m_condensed = m – mx₂ = (1 – x₂) m = ( 1 – 0.3000 ) ( 68.222 lb_m ) = 47.75 lb_m.

We have to find the internal energy at the initial and final states to compute the heat transfer.  The initial internal energy is simply u_g(160 psia) = 108.50 Btu/lb_m (Table A-12E).  The internal energy at the final state is found from the saturation properties and the quality, x₂ = 0.3000.

The heat transfer can now be found from the first law.

Q = DU + W = m(u₂ – u₁) + W = (68.222 lb_m)(47.396 – 108.50) Btu/lb_m + 0 = –4 ,169 Btu.

4    An insulated piston-cylinder device contains 5 L of saturated liquid water at a constant pressure of 175 kPa.  Water is stirred by a paddle wheel while a current of 8 A flows for 45 min through a resistor placed in the water.  If one half of the liquid is evaporated during this constant-pressure process and the paddle-wheel work amounts to 400 kJ, determine the voltage of the source.  Also, show the process on a P-v diagram with respect to the saturation lines.

In this problem we assume that the insulation on the piston-cylinder device reduces the heat transfer to a negligible amount so that we may assume that Q = 0.  The evaporation of the liquid comes from the addition of work from the resistor and the paddle wheel.  At the same time, the piston is expanding, at constant pressure, so that the water is doing work.  Since the pressure is constant, this work is simply equal to P(V₂ – V₁).

For this problem the first law becomes:

Q = 0 = DU + W = m(u₂ – u₁) + P(V₂ – V₁) + W_resistor + W_paddle

We can write the volumes as the product of mass times specific volume and use the fact that P = P₁ = P₂ to introduce the enthalpy, h = u + Pv.

–W_resistor – W_paddle = m(u₂ – u₁) + Pm(v₂ – v₁) =m[(u₂ + P₂v₂) – (u₁ + P₁v₁)] = m(h₂ – h₁)

We can find the mass from the initial state where we know V₁ = 5 L and v₁ = v_f(P₁ = 175 kPa).  Using Table A-5 on page 916 to find v_f, we compute the mass as follows.

Since the paddle wheel and resistance work are inputs, these are negative.  Thus the values of W_resistor and W_paddle are –EIDt and –400 kJ, where E is the voltage that we want to find and I is the given current of 8 A.  The enthalpy at the initial state is simply h_f at 175 kPa.  The enthalpy at the final state is the enthalpy of a mixture with a quality of 50% at the same 175 kPa pressure.  We thus have h₁ = h_f(175 kPa) = 487.01 kJ/kg, where we use Table A-5 for the saturation data.  Using the same table and the value of x₂ = 50% we find h₂ as follows.

Substituting the work terms and enthalpy and mass into our first law gives an equation for the unknown voltage.

–W_resistor – W_paddle = EIDt + 400 kJ = m(h₂ – h₁) = (4.730 kg)(1593.56 – 487.01) kJ/kg

Combining the numerical data gives an equation for the work done by the resistor.

EIDt = 4834 kJ = 4834 kW•s

We can solve this to find the voltage applied to the resistor.

5    A piston-cylinder device initially contains steam at 200 kPa, 200^oC, and 0.5 m³.  At this state a linear spring (F a x) is touching the piston but exerts no force on it.  Heat is now slowly transferred to the steam causing the pressure and the volume to rise to 500 kPa and 0.6 m³, respectively.  Show the process on a P-v diagram with respect to the saturation lines and determine (a) the final temperature, (b) the work done by the steam, and (c) the total heat transferred.

The path for this process is shown in the diagram below.  The initial point of the path is (P₁, V₁) = (200 kPa, 0.5 m³); the final point is (P₂, V₂) = (500 kPa, 0.6 m³).

                   

 The linear path results from the linear relationship between spring force and displacement and the linear relationship between displacement and volume.  The equilibrium point of the spring occurs at the initial volume, since the spring is touching the piston but exerting no force at this point.  The pressure due to the spring is then given by the following equation.

The total pressure, P, acting on the water will be the sum of the constant pressure from the weight of the piston (and atmospheric pressure), P₁, and P_spring. 

This equation is seen to provide a linear relationship between pressure and volume as shown in the diagram above.  The work is the area under the path which is the area of a trapezoid: W = (P₁ + P₂)(V₂ – V₁)/2.  Using the values of pressure and volume given in the problem allows us to find the work.

The heat transfer is given by the first law: Q = DU + W = m(u₂ – u₁) + W.  We have to find the values of u₁ and u₂ from the property tables for water.  In addition, we can find the mass from the initial volume, V₁ = 0.5 m³, and the initial specific volume, v₁.

At the initial state of P₁ = 200 kPa and T₁ = 200^oC, we find the following properties in the superheat table, Table A-6 on page 918: v₁ = 1.08049 m³/kg and u₁ = 2654.6 kJ/kg.  We can then find the mass as follows.

From this system mass, we can compute the specific volume at the final state where V₂ = 0.6 m³.

At the final pressure of 500 kPa, the specific volume, v₂ = 1.2966 m³/kg occurs between temperatures of 1100^oC and 1200^oC in Table A-6 on page 920.  Interpolating between these two points gives the temperature and internal energy at the final state.

`

So, the final temperature, T₂ = 1132^oC.

The heat transfer is found from the first law as usual.

Q = m(u₂ – u₁) + W = (0.46275 kg)(4325.8 – 2654.6)kJ/kg + 35 kJ = 808 kJ.

6   6.A piston-cylinder device initially contains 0.8 m³ of saturated water vapor at 250 kPa.  At this state, the piston is resting on a set of stops and the mass of the piston is such that a pressure of 300 kPa is required to move it.  Heat is now slowly transferred to the steam until the volume doubles.  Show the process on a P-V diagram with respect to saturation lines and determine (a) the final temperature, (b) the work done during this process, and (c) the total heat transfer.

The P-v diagram for this process is shown at the right.  During the first part of the process, from point 1 to point 2, the pressure is too low to lift the piston, so the process occurs at constant volume.  Once the pressure reaches 300 kPa, the piston starts to rise and the remainder of the expansion (a doubling of volume) occurs at constant pressure.

The total heat transfer is found from the first law as Q = DU + W = m(u₃ – u₁) + W, where W is found as the area under the path: W = P_2-3(V₃ – V₁) and P_2-3 is the constant pressure of 300 kPa = P₂ = P₃.  We know that V₁ = 0.8 m³ and V₃ = 2 V₁ = 1.6 m³.  Thus, the work is found as follows.

The mass is found from the initial volume of 0.8 m³ and the initial specific volume which is the specific volume of a saturated vapor at 250 kPa.  This is found from Table A-5 on page 916.

Since the mass is constant and the volume doubles, the final specific volume is twice the initial specific volume: v₃ = 2v₂ = 1.43746 m³/kg.  The final state with this specific volume and a pressure of 300 kPa (0.3 MPa) is found in the superheat tables, Table A-6 on page 918, to occur at a temperature between 600^oC and 700^oC.  The final temperature is found by interpolation.

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So the final temperature, T₃ = 662^oC.

The initial internal energy for the saturated vapor state is found from Table A-5, u₁ = u_g(250 kPa) = 2536.8 kJ/kg.  The internal energy at the final state is found by an interpolation similar to the one used for the temperature.

The heat transfer is found from the usual equation for the first law.

Q = m(u₃ – u₁) + W = (1.113 kg)(3412.3 – 2536.8)kJ/kg + 240 kJ = 1214 kJ.