Energy Balances on Unit Operations and Equipment

First-law bookkeeping across heaters, exchangers, turbines and pumps requires temperature-dependent enthalpy when variation is significant and consistent sign conventions.

Core learning goals

• Explain and apply Enthalpy references and sensible heat, including assumptions and independent verification.
• Explain and apply Nonreactive process heat balance, including assumptions and independent verification.
• Explain and apply Heat exchanger effectiveness, including assumptions and independent verification.
• Explain and apply Pumps, compressors and flow work, including assumptions and independent verification.

Lessons and reading order

• [Lesson 6.1: Enthalpy references and sensible heat](../lessons/unit-06-lesson-01.html)
• [Lesson 6.2: Nonreactive process heat balance](../lessons/unit-06-lesson-02.html)
• [Lesson 6.3: Heat exchanger effectiveness](../lessons/unit-06-lesson-03.html)
• [Lesson 6.4: Pumps, compressors and flow work](../lessons/unit-06-lesson-04.html)

Unit synthesis: identify what is known

First-law bookkeeping across heaters, exchangers, turbines and pumps requires temperature-dependent enthalpy when variation is significant and consistent sign conventions. A strong solution begins with a process sketch and a stream table. Write values to a stated basis, mark all missing data, and calculate degrees of freedom before attempting numerical iteration. A flow diagram is useful only if the stream symbols and assumptions can be traced into actual balance equations. If an unknown variable is introduced to close a balance, explain what physical instrument or property measurement would establish it.

Connecting the four lessons

The first lesson establishes a local quantity or concept, while the later lessons expose how its interpretation changes with additional process complexity. For example, mass fractions remain valid for a blend but reacting compositions need generation terms; heat duties may be linear in temperature under constant heat capacity but nonlinear near a phase transition. Develop a single reference stream table and compare how each analytical step changes when more units are added.

Diagnostic exercise

Assume an otherwise plausible design produces an impossible negative outlet or a reported overall recovery of 105%. Instead of clipping the number, trace which equation or boundary was omitted or whose units were inconsistent. Check whether the solver was overdetermined by a redundant balance or underdetermined by a missing composition. Document the correction and rerun your limiting-case tests.

A second diagnostic asks you to separate measured and assumed quantities. Label uncertainties if known; do not attach precision to idealized coefficients. A balance audit should preserve the original measurements and record any reconciliation separately. This habit will become important in the integrated case studies and final project.

Unit completion exercise

Draw one example flowsheet for this unit and formulate an overall balance plus two meaningful independent additional equations, if the process contains enough components. State why each equation is independent. Determine what unknowns remain and choose the cheapest credible measurement to close the problem. Explain whether the modified process is still steady, and assess one alternative operating condition without assuming unchanged performance.

Unit 6.1: Enthalpy references and sensible heat

Book 2: Material and Energy Balances — Energy balances on process equipment

What you will be able to do

Enthalpy differences are physical; absolute reference choices cancel in a consistent balance. Interpret Delta_h=int_T1^T2 Cp(T)dT; for constant Cp, Delta_h=Cp*(T2-T1) and demonstrate how it follows from an appropriate boundary. Identify whether the equation describes an overall, species, elemental, or energy balance; discuss what is actually measured and the conditions under which the balance is valid.

Essential physical insight

Sizing a jacket for a recirculating heat-transfer fluid. The critical objective is not to memorize a plug-in formula but to recognize conserved quantities and define the physical process correctly. Break the problem into stream variables, known specifications, unknowns, and constitutive or property relationships. A numerical estimate is an engineering claim only if its basis and limitations are reproducible.

Deriving the balance from the process boundary

Start from the observations implied by enthalpy references and sensible heat. Enthalpy differences are physical; absolute reference choices cancel in a consistent balance. Represent each material crossing the boundary with a signed rate and state its measurement basis. If the equipment retains material, include an accumulation term. A balance written per second cannot be directly combined with a plant log written per shift without converting the interval; similarly, mass fractions and molar compositions cannot be used in the same species account without a molecular-weight transformation. Before any algebra, list feed states, outlet states, any generation or consumption, and the measured inventory change.

Write the central relationship as Delta_h=int_T1^T2 Cp(T)dT; for constant Cp, Delta_h=Cp*(T2-T1). Read every variable aloud with its units, explaining which terms are independent measurements and which are derived. For a process with reaction, the component generation term is not optional. For a nonreactive mixer, a nonzero computed generation indicates a modeling or measurement mistake. Use unit consistency as the first diagnostic check, and then compare the predicted result against simple physical limits.

A worked calculation and its interpretation

Illustrative case. Oil at 2 kg/s and cp=2.2 kJ/(kg K) heating by 45 K needs 198 kW, excluding loss. This numerical calculation is a worked illustration and not a measurement from an operating facility. Write the stream table with at least one named component, a physical total rate, and a units column. Substitute numbers into the governing equation and solve the unknown before rounding. Keep at least one more significant figure in intermediate calculations than will be reported to the operator.

The numerical result has value only within the stated assumptions. Combining enthalpy values with inconsistent reference temperatures produces spurious duties. Confirm that any calculated component fraction is between zero and one, that recoveries never exceed their admissible range and that energy duties obey the stated sign convention. A unit conversion error can shift a prediction by factors of 1,000 or 3,600 even when the arithmetic appears internally consistent.

Why the answer changes in a real plant

Consider the industrial case: Sizing a jacket for a recirculating heat-transfer fluid. Before trusting a software result, identify temperature and pressure effects, instrument calibration, inventory fluctuations and unrecorded process streams. A perfectly balanced flowsheet can still be a poor representation of reality if one instrument reads systematically low. Conversely, a balance residual does not by itself prove a physical leak or chemical loss.

To decide whether further detail is necessary, first quantify the decision at issue. A rough balance may be sufficient to choose between two pilot runs but not to specify relief equipment, guarantee product quality, or assert environmental compliance. Explain which property or stream measurement would most reduce uncertainty, and how it could be obtained without introducing a new uncontrolled hazard.

Guided redesign exercise

Imagine a proposed 20% increase in the process feed or load. Identify which terms in Delta_h=int_T1^T2 Cp(T)dT; for constant Cp, Delta_h=Cp*(T2-T1) scale linearly and which could change because of capacity limits, residence time, equilibrium, or heat transfer. Recompute the balance on a consistent time basis and explain why unchanged efficiency or composition may be an unreasonable assumption. Do not silently extrapolate a laboratory coefficient outside the range where it was measured.

Prepare a second scenario in which the least certain input changes by 10% while the other stated conditions remain fixed. Estimate whether the output changes proportionally. If the result violates a conservation limit, retrace variable definitions. If it remains feasible, identify an observation that could distinguish between the original and changed scenarios.

Verification and critical-thinking questions

1. State the system boundary and draw the streams for enthalpy references and sensible heat; label every unknown rather than assigning guessed values.
2. Explain the difference between an identity enforced by conservation and an empirical correlation that may fail at changed conditions.
3. What measurement would most directly test the assumption: enthalpy differences are physical; absolute reference choices cancel in a consistent balance?
4. Recompute the illustrative case using a different but consistent calculation basis. Which outputs must remain invariant?
5. Construct a limiting-case test and explain precisely what would constitute a physically impossible result.

Practical check: Combining enthalpy values with inconsistent reference temperatures produces spurious duties. A satisfactory answer should separate the equation, numerical result, process interpretation, and limitations rather than ending with a number alone.

Practical laboratory connection

The laboratory models in this volume can illustrate how a calculated response changes as conditions vary. Locate a related simulation in the book index, write down a prediction before operating its controls, and test whether the result is monotone, bounded or approaching an asymptotic limit. Compare at least one displayed numerical result with an independent calculation. A graphical match is not sufficient: test numerical values and units separately.

Further study

Consult the freely accessible LearnChemE Material and Energy Balances resources and the NIST Chemistry WebBook for property-data methodology. Use a suitable authoritative property model for real design rather than applying constant heat capacities or ideal-mixture assumptions indiscriminately.

Unit 6.2: Nonreactive process heat balance

Book 2: Material and Energy Balances — Energy balances on process equipment

What you will be able to do

Steady open-system balance includes heat, work and kinetic and potential changes. Interpret Qdot-Wdot=sum(out mh)-sum(in mh)+Delta(KE+PE) and demonstrate how it follows from an appropriate boundary. Identify whether the equation describes an overall, species, elemental, or energy balance; discuss what is actually measured and the conditions under which the balance is valid.

Essential physical insight

Determining electrical-heater power from a temperature rise. The critical objective is not to memorize a plug-in formula but to recognize conserved quantities and define the physical process correctly. Break the problem into stream variables, known specifications, unknowns, and constitutive or property relationships. A numerical estimate is an engineering claim only if its basis and limitations are reproducible.

Deriving the balance from the process boundary

Start from the observations implied by nonreactive process heat balance. Steady open-system balance includes heat, work and kinetic and potential changes. Represent each material crossing the boundary with a signed rate and state its measurement basis. If the equipment retains material, include an accumulation term. A balance written per second cannot be directly combined with a plant log written per shift without converting the interval; similarly, mass fractions and molar compositions cannot be used in the same species account without a molecular-weight transformation. Before any algebra, list feed states, outlet states, any generation or consumption, and the measured inventory change.

Write the central relationship as Qdot-Wdot=sum(out mh)-sum(in mh)+Delta(KE+PE). Read every variable aloud with its units, explaining which terms are independent measurements and which are derived. For a process with reaction, the component generation term is not optional. For a nonreactive mixer, a nonzero computed generation indicates a modeling or measurement mistake. Use measurable stream composition as the first diagnostic check, and then compare the predicted result against simple physical limits.

A worked calculation and its interpretation

Illustrative case. A 0.8 kg/s stream heated 30 K at cp=3.5 kJ/(kg K) needs 84 kW if shaft work and losses are negligible. This numerical calculation is a worked illustration and not a measurement from an operating facility. Write the stream table with at least one named component, a physical total rate, and a units column. Substitute numbers into the governing equation and solve the unknown before rounding. Keep at least one more significant figure in intermediate calculations than will be reported to the operator.

The numerical result has value only within the stated assumptions. Identify the sign convention and avoid counting shaft work as heat. Confirm that any calculated component fraction is between zero and one, that recoveries never exceed their admissible range and that energy duties obey the stated sign convention. A unit conversion error can shift a prediction by factors of 1,000 or 3,600 even when the arithmetic appears internally consistent.

Why the answer changes in a real plant

Consider the industrial case: Determining electrical-heater power from a temperature rise. Before trusting a software result, identify temperature and pressure effects, instrument calibration, inventory fluctuations and unrecorded process streams. A perfectly balanced flowsheet can still be a poor representation of reality if one instrument reads systematically low. Conversely, a balance residual does not by itself prove a physical leak or chemical loss.

To decide whether further detail is necessary, first quantify the decision at issue. A rough balance may be sufficient to choose between two pilot runs but not to specify relief equipment, guarantee product quality, or assert environmental compliance. Explain which property or stream measurement would most reduce uncertainty, and how it could be obtained without introducing a new uncontrolled hazard.

Guided redesign exercise

Imagine a proposed 20% increase in the process feed or load. Identify which terms in Qdot-Wdot=sum(out mh)-sum(in mh)+Delta(KE+PE) scale linearly and which could change because of capacity limits, residence time, equilibrium, or heat transfer. Recompute the balance on a consistent time basis and explain why unchanged efficiency or composition may be an unreasonable assumption. Do not silently extrapolate a laboratory coefficient outside the range where it was measured.

Prepare a second scenario in which the least certain input changes by 10% while the other stated conditions remain fixed. Estimate whether the output changes proportionally. If the result violates a conservation limit, retrace variable definitions. If it remains feasible, identify an observation that could distinguish between the original and changed scenarios.

Verification and critical-thinking questions

1. State the system boundary and draw the streams for nonreactive process heat balance; label every unknown rather than assigning guessed values.
2. Explain the difference between an identity enforced by conservation and an empirical correlation that may fail at changed conditions.
3. What measurement would most directly test the assumption: steady open-system balance includes heat, work and kinetic and potential changes?
4. Recompute the illustrative case using a different but consistent calculation basis. Which outputs must remain invariant?
5. Construct a limiting-case test and explain precisely what would constitute a physically impossible result.

Practical check: Identify the sign convention and avoid counting shaft work as heat. A satisfactory answer should separate the equation, numerical result, process interpretation, and limitations rather than ending with a number alone.

Practical laboratory connection

The laboratory models in this volume can illustrate how a calculated response changes as conditions vary. Locate a related simulation in the book index, write down a prediction before operating its controls, and test whether the result is monotone, bounded or approaching an asymptotic limit. Compare at least one displayed numerical result with an independent calculation. A graphical match is not sufficient: test numerical values and units separately.

Further study

Consult the freely accessible LearnChemE Material and Energy Balances resources and the NIST Chemistry WebBook for property-data methodology. Use a suitable authoritative property model for real design rather than applying constant heat capacities or ideal-mixture assumptions indiscriminately.

Unit 6.3: Heat exchanger effectiveness

Book 2: Material and Energy Balances — Energy balances on process equipment

What you will be able to do

Heat exchanger duty is bounded by the minimum stream heat capacity. Interpret Q=epsilonC_min(T_h,in-T_c,in); NTU=UA/C_min and demonstrate how it follows from an appropriate boundary. Identify whether the equation describes an overall, species, elemental, or energy balance; discuss what is actually measured and the conditions under which the balance is valid.

Essential physical insight

Plate heat-exchanger performance across seasonal feed conditions. The critical objective is not to memorize a plug-in formula but to recognize conserved quantities and define the physical process correctly. Break the problem into stream variables, known specifications, unknowns, and constitutive or property relationships. A numerical estimate is an engineering claim only if its basis and limitations are reproducible.

Deriving the balance from the process boundary

Start from the observations implied by heat exchanger effectiveness. Heat exchanger duty is bounded by the minimum stream heat capacity. Represent each material crossing the boundary with a signed rate and state its measurement basis. If the equipment retains material, include an accumulation term. A balance written per second cannot be directly combined with a plant log written per shift without converting the interval; similarly, mass fractions and molar compositions cannot be used in the same species account without a molecular-weight transformation. Before any algebra, list feed states, outlet states, any generation or consumption, and the measured inventory change.

Write the central relationship as Q=epsilonC_min(T_h,in-T_c,in); NTU=UA/C_min. Read every variable aloud with its units, explaining which terms are independent measurements and which are derived. For a process with reaction, the component generation term is not optional. For a nonreactive mixer, a nonzero computed generation indicates a modeling or measurement mistake. Use physical limit cases as the first diagnostic check, and then compare the predicted result against simple physical limits.

A worked calculation and its interpretation

Illustrative case. If hot capacity rate is 6 kW/K and cold is 4 kW/K, ΔT_in=50 K and effectiveness 0.7, Q=140 kW. This numerical calculation is a worked illustration and not a measurement from an operating facility. Write the stream table with at least one named component, a physical total rate, and a units column. Substitute numbers into the governing equation and solve the unknown before rounding. Keep at least one more significant figure in intermediate calculations than will be reported to the operator.

The numerical result has value only within the stated assumptions. A calculated outlet crossing the thermodynamic maximum indicates a sign, unit or model error. Confirm that any calculated component fraction is between zero and one, that recoveries never exceed their admissible range and that energy duties obey the stated sign convention. A unit conversion error can shift a prediction by factors of 1,000 or 3,600 even when the arithmetic appears internally consistent.

Why the answer changes in a real plant

Consider the industrial case: Plate heat-exchanger performance across seasonal feed conditions. Before trusting a software result, identify temperature and pressure effects, instrument calibration, inventory fluctuations and unrecorded process streams. A perfectly balanced flowsheet can still be a poor representation of reality if one instrument reads systematically low. Conversely, a balance residual does not by itself prove a physical leak or chemical loss.

To decide whether further detail is necessary, first quantify the decision at issue. A rough balance may be sufficient to choose between two pilot runs but not to specify relief equipment, guarantee product quality, or assert environmental compliance. Explain which property or stream measurement would most reduce uncertainty, and how it could be obtained without introducing a new uncontrolled hazard.

Guided redesign exercise

Imagine a proposed 20% increase in the process feed or load. Identify which terms in Q=epsilonC_min(T_h,in-T_c,in); NTU=UA/C_min scale linearly and which could change because of capacity limits, residence time, equilibrium, or heat transfer. Recompute the balance on a consistent time basis and explain why unchanged efficiency or composition may be an unreasonable assumption. Do not silently extrapolate a laboratory coefficient outside the range where it was measured.

Prepare a second scenario in which the least certain input changes by 10% while the other stated conditions remain fixed. Estimate whether the output changes proportionally. If the result violates a conservation limit, retrace variable definitions. If it remains feasible, identify an observation that could distinguish between the original and changed scenarios.

Verification and critical-thinking questions

1. State the system boundary and draw the streams for heat exchanger effectiveness; label every unknown rather than assigning guessed values.
2. Explain the difference between an identity enforced by conservation and an empirical correlation that may fail at changed conditions.
3. What measurement would most directly test the assumption: heat exchanger duty is bounded by the minimum stream heat capacity?
4. Recompute the illustrative case using a different but consistent calculation basis. Which outputs must remain invariant?
5. Construct a limiting-case test and explain precisely what would constitute a physically impossible result.

Practical check: A calculated outlet crossing the thermodynamic maximum indicates a sign, unit or model error. A satisfactory answer should separate the equation, numerical result, process interpretation, and limitations rather than ending with a number alone.

Practical laboratory connection

The laboratory models in this volume can illustrate how a calculated response changes as conditions vary. Locate a related simulation in the book index, write down a prediction before operating its controls, and test whether the result is monotone, bounded or approaching an asymptotic limit. Compare at least one displayed numerical result with an independent calculation. A graphical match is not sufficient: test numerical values and units separately.

Further study

Consult the freely accessible LearnChemE Material and Energy Balances resources and the NIST Chemistry WebBook for property-data methodology. Use a suitable authoritative property model for real design rather than applying constant heat capacities or ideal-mixture assumptions indiscriminately.

Unit 6.4: Pumps, compressors and flow work

Book 2: Material and Energy Balances — Energy balances on process equipment

What you will be able to do

Shaft power and fluid enthalpy increase are linked by efficiency and flow work. Interpret P_hydraulic=rhogQ_vol*H; P_shaft=P_hydraulic/eta and demonstrate how it follows from an appropriate boundary. Identify whether the equation describes an overall, species, elemental, or energy balance; discuss what is actually measured and the conditions under which the balance is valid.

Essential physical insight

Pump retrofit with measured head, flow and electrical power. The critical objective is not to memorize a plug-in formula but to recognize conserved quantities and define the physical process correctly. Break the problem into stream variables, known specifications, unknowns, and constitutive or property relationships. A numerical estimate is an engineering claim only if its basis and limitations are reproducible.

Deriving the balance from the process boundary

Start from the observations implied by pumps, compressors and flow work. Shaft power and fluid enthalpy increase are linked by efficiency and flow work. Represent each material crossing the boundary with a signed rate and state its measurement basis. If the equipment retains material, include an accumulation term. A balance written per second cannot be directly combined with a plant log written per shift without converting the interval; similarly, mass fractions and molar compositions cannot be used in the same species account without a molecular-weight transformation. Before any algebra, list feed states, outlet states, any generation or consumption, and the measured inventory change.

Write the central relationship as P_hydraulic=rhogQ_vol*H; P_shaft=P_hydraulic/eta. Read every variable aloud with its units, explaining which terms are independent measurements and which are derived. For a process with reaction, the component generation term is not optional. For a nonreactive mixer, a nonzero computed generation indicates a modeling or measurement mistake. Use independent balance closure as the first diagnostic check, and then compare the predicted result against simple physical limits.

A worked calculation and its interpretation

Illustrative case. For water at 1000 kg/m^3, 0.02 m^3/s and 30 m head, power delivered to water is 5.886 kW; at efficiency 0.75 input is 7.848 kW. This numerical calculation is a worked illustration and not a measurement from an operating facility. Write the stream table with at least one named component, a physical total rate, and a units column. Substitute numbers into the governing equation and solve the unknown before rounding. Keep at least one more significant figure in intermediate calculations than will be reported to the operator.

The numerical result has value only within the stated assumptions. Do not divide by efficiency when calculating useful fluid power. Confirm that any calculated component fraction is between zero and one, that recoveries never exceed their admissible range and that energy duties obey the stated sign convention. A unit conversion error can shift a prediction by factors of 1,000 or 3,600 even when the arithmetic appears internally consistent.

Why the answer changes in a real plant

Consider the industrial case: Pump retrofit with measured head, flow and electrical power. Before trusting a software result, identify temperature and pressure effects, instrument calibration, inventory fluctuations and unrecorded process streams. A perfectly balanced flowsheet can still be a poor representation of reality if one instrument reads systematically low. Conversely, a balance residual does not by itself prove a physical leak or chemical loss.

To decide whether further detail is necessary, first quantify the decision at issue. A rough balance may be sufficient to choose between two pilot runs but not to specify relief equipment, guarantee product quality, or assert environmental compliance. Explain which property or stream measurement would most reduce uncertainty, and how it could be obtained without introducing a new uncontrolled hazard.

Guided redesign exercise

Imagine a proposed 20% increase in the process feed or load. Identify which terms in P_hydraulic=rhogQ_vol*H; P_shaft=P_hydraulic/eta scale linearly and which could change because of capacity limits, residence time, equilibrium, or heat transfer. Recompute the balance on a consistent time basis and explain why unchanged efficiency or composition may be an unreasonable assumption. Do not silently extrapolate a laboratory coefficient outside the range where it was measured.

Prepare a second scenario in which the least certain input changes by 10% while the other stated conditions remain fixed. Estimate whether the output changes proportionally. If the result violates a conservation limit, retrace variable definitions. If it remains feasible, identify an observation that could distinguish between the original and changed scenarios.

Verification and critical-thinking questions

1. State the system boundary and draw the streams for pumps, compressors and flow work; label every unknown rather than assigning guessed values.
2. Explain the difference between an identity enforced by conservation and an empirical correlation that may fail at changed conditions.
3. What measurement would most directly test the assumption: shaft power and fluid enthalpy increase are linked by efficiency and flow work?
4. Recompute the illustrative case using a different but consistent calculation basis. Which outputs must remain invariant?
5. Construct a limiting-case test and explain precisely what would constitute a physically impossible result.

Practical check: Do not divide by efficiency when calculating useful fluid power. A satisfactory answer should separate the equation, numerical result, process interpretation, and limitations rather than ending with a number alone.

Practical laboratory connection

The laboratory models in this volume can illustrate how a calculated response changes as conditions vary. Locate a related simulation in the book index, write down a prediction before operating its controls, and test whether the result is monotone, bounded or approaching an asymptotic limit. Compare at least one displayed numerical result with an independent calculation. A graphical match is not sufficient: test numerical values and units separately.

Further study

Consult the freely accessible LearnChemE Material and Energy Balances resources and the NIST Chemistry WebBook for property-data methodology. Use a suitable authoritative property model for real design rather than applying constant heat capacities or ideal-mixture assumptions indiscriminately.

Applied project 6: District heat-exchanger network and temperature approach

Scenario and process objective

A hot process stream enters a countercurrent exchanger at 130°C with heat-capacity rate 7 kW/K. The cold stream enters at 30°C with heat-capacity rate 4 kW/K. A compact exchanger provides UA=5 kW/K. The manager wants to recover more heat but proposes only increasing hot-side flow, ignoring the cold-side capacity and minimum-approach constraints.

The project must be documented as a sequence of claims: identify what conservation enforces, which property data are assumed, which quantities are operator observations, and which decisions remain uncertain. Prepare a separate stream table for every external feed or outlet. Do not insert convenient values for missing streams and then present the case as uniquely solved. A correctly identified underdetermined model is more valuable than an unjustified numerical answer.

Calculation and reference result

Here Cmin=4, Cmax=7, Cr≈0.5714 and NTU=5/4=1.25. Countercurrent effectiveness is (1−exp[−1.25(1−0.5714)])/(1−0.5714 exp[−1.25(1−0.5714)])≈0.620. The screening heat duty is therefore roughly 248 kW. Cold outlet is approximately 92°C and hot outlet approximately 94.6°C.

The figures above are idealized training values; they are not operational measurements. Repeat the calculation independently using base SI units and show at least one back-substitution into the original overall or species balance. Keep dimensional conversions visible, especially if a heat rate passes between kJ/h and kW or a composition passes between percent and fraction.

Model criticism and operating context

The model assumes constant properties, no heat loss, no phase change and a valid countercurrent arrangement. Increasing hot-side capacity rate is not the same as raising UA and may produce diminishing returns. Check every outlet against energy conservation and consider realistic fouling and pressure drop.

A design report needs a statement of operating range and expected effects that the simplified equations leave out. For example, thermodynamic equilibrium may be inappropriate during a rapid transient; a nominal heat-transfer coefficient may depend on fouling or regime; a flow meter may give an average that is misaligned with an inventory sampling time. Identify which omitted effect could reverse the preferred operating decision.

Required investigation

Explore the impact of reducing UA by 30% to represent fouling. Explain the energy and economic tradeoff between heat-exchanger cleaning and added steam, without inventing fuel tariffs.

Provide a graphical sensitivity study with clearly labeled axes and units. Use at least two independent checks on the reported values. One check must be a conservation identity and the other a dimensional, physical-bounds, or alternative-method check. If the case is not fully determined, specify the least expensive measurement that would make the next calculation defensible, and quantify how measurement uncertainty could influence the decision.

Extension: implementation and quality assurance

Design a two-stage implementation plan: first, a low-cost experiment or audit, and second, a revised model calibrated with those observations. Decide how you would document instrumentation, calibration, sampling and data provenance. Provide a concise statement of operational risks that would prevent deployment of the model without expert supervision.

Decision memo template

• What is known: state the measured feed and product values, date range, units and calculation basis.
• What is calculated: identify conserved totals, composition conversions and any heat duty.
• What is assumed: list ideal-gas behavior, perfect mixing, constant heat capacity, reaction completion or transfer efficiency only if used.
• What remains uncertain: name unmeasured streams, material properties or error correlations.
• What to do next: propose a specific measurement and the change in decision it could permit.

Questions for viva or peer discussion

1. Which equation would become invalid if inventory changed over the measurement window?
2. Can a numerically closed overall mass balance conceal a large component imbalance? Construct an example.
3. What is the most dangerous units mistake for this case, and what magnitude would it introduce?
4. Explain the effect of a 10% feed increase on both production and utility rates, including capacity limits.
5. What evidence would allow a simplified teaching model to support a plant engineering decision?

Assessment — Unit 6: Energy balances on process equipment

These questions demand a calculation or a defended explanation. State units and assumptions for every numerical answer.

Topic 6.1: Enthalpy references and sensible heat

1. Derivation. Build a balance for: Sizing a jacket for a recirculating heat-transfer fluid. Derive the expression `Delta_h=int_T1^T2 Cp(T)dT; for constant Cp, Delta_h=Cp*(T2-T1)` or explain why an additional relationship is necessary.

2. Quantitative check. Recalculate this illustrative scenario without quoting the reported result: Oil at 2 kg/s and cp=2.2 kJ/(kg K) heating by 45 K needs 198 kW, excluding loss. Record a self-check based on conservation.

3. Error diagnosis. A colleague proposes an answer that contradicts this principle: Combining enthalpy values with inconsistent reference temperatures produces spurious duties. Identify the failed assumption and describe a corrective measurement.

4. Sensitivity. Change the most influential given input by +10%, leaving other inputs fixed. Determine the new result where the data suffice; otherwise explicitly name the missing information.

5. Boundary test. Present a physically meaningful limit and show how the predicted result behaves. State whether a thermodynamic property correlation remains valid in that limit.

Topic 6.2: Nonreactive process heat balance

1. Derivation. Build a balance for: Determining electrical-heater power from a temperature rise. Derive the expression `Qdot-Wdot=sum(out m*h)-sum(in m*h)+Delta(KE+PE)` or explain why an additional relationship is necessary.

2. Quantitative check. Recalculate this illustrative scenario without quoting the reported result: A 0.8 kg/s stream heated 30 K at cp=3.5 kJ/(kg K) needs 84 kW if shaft work and losses are negligible. Record a self-check based on conservation.

3. Error diagnosis. A colleague proposes an answer that contradicts this principle: Identify the sign convention and avoid counting shaft work as heat. Identify the failed assumption and describe a corrective measurement.

4. Sensitivity. Change the most influential given input by +10%, leaving other inputs fixed. Determine the new result where the data suffice; otherwise explicitly name the missing information.

5. Boundary test. Present a physically meaningful limit and show how the predicted result behaves. State whether a thermodynamic property correlation remains valid in that limit.

Topic 6.3: Heat exchanger effectiveness

1. Derivation. Build a balance for: Plate heat-exchanger performance across seasonal feed conditions. Derive the expression `Q=epsilon*C_min*(T_h,in-T_c,in); NTU=UA/C_min` or explain why an additional relationship is necessary.

2. Quantitative check. Recalculate this illustrative scenario without quoting the reported result: If hot capacity rate is 6 kW/K and cold is 4 kW/K, ΔT_in=50 K and effectiveness 0.7, Q=140 kW. Record a self-check based on conservation.

3. Error diagnosis. A colleague proposes an answer that contradicts this principle: A calculated outlet crossing the thermodynamic maximum indicates a sign, unit or model error. Identify the failed assumption and describe a corrective measurement.

4. Sensitivity. Change the most influential given input by +10%, leaving other inputs fixed. Determine the new result where the data suffice; otherwise explicitly name the missing information.

5. Boundary test. Present a physically meaningful limit and show how the predicted result behaves. State whether a thermodynamic property correlation remains valid in that limit.

Topic 6.4: Pumps, compressors and flow work

1. Derivation. Build a balance for: Pump retrofit with measured head, flow and electrical power. Derive the expression `P_hydraulic=rho*g*Q_vol*H; P_shaft=P_hydraulic/eta` or explain why an additional relationship is necessary.

2. Quantitative check. Recalculate this illustrative scenario without quoting the reported result: For water at 1000 kg/m^3, 0.02 m^3/s and 30 m head, power delivered to water is 5.886 kW; at efficiency 0.75 input is 7.848 kW. Record a self-check based on conservation.

3. Error diagnosis. A colleague proposes an answer that contradicts this principle: Do not divide by efficiency when calculating useful fluid power. Identify the failed assumption and describe a corrective measurement.

4. Sensitivity. Change the most influential given input by +10%, leaving other inputs fixed. Determine the new result where the data suffice; otherwise explicitly name the missing information.

5. Boundary test. Present a physically meaningful limit and show how the predicted result behaves. State whether a thermodynamic property correlation remains valid in that limit.

Grading rubric

Award credit for a correctly defined boundary and flow basis (25%), an equation with independent unknowns (25%), numerical units and magnitude checks (20%), critical assumptions and uncertainty (20%), and readable presentation (10%). More digits are not a substitute for property data.