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Decision Lab · E3 · Arrangement, capacity rate and NTU

Unknown outlet temperatures: choose the exchanger model

Choose a method before assuming an outlet temperature. Then test whether changing the flow arrangement preserves the result.

Original case · Suggested 20 minutes · Edition 2026-10-12.1

By the end of this workshop

  • Choose a method that respects the supplied conditions.
  • Explain how a changed condition affects the decision.
  • Identify a physical or procedural cross-check and an applicability limit.
  • Record the assumption you would verify next time.

Start with the complete model

Givens and assumptions

  • Steady sensible exchanger, no external heat loss; constant properties.
  • Hot inlet 160°F, hot capacity rate 2,400 Btu/(h·°F). Cold inlet 60°F, cold capacity rate 1,200 Btu/(h·°F).
  • UA = 1,800 Btu/(h·°F); counterflow arrangement. Both outlet temperatures are unknown.
  • Use the supplied relationships below. Capacity rates and UA keep the same units.

Supplied relationships

Use the appropriate arrangement with the stated capacity rates and inlet conditions.

Cmin=min⁡(Ch,Cc),Cr=Cmin/Cmax,NTU=UA/CminC_{min}=\min(C_h,C_c),\quad C_r=C_{min}/C_{max},\quad NTU=UA/C_{min}Q˙max=Cmin(Th,i−Tc,i),Q˙=εQ˙max\dot Q_{max}=C_{min}(T_{h,i}-T_{c,i}),\quad\dot Q=\varepsilon\dot Q_{max}εcf=1−e−NTU(1−Cr)1−Cre−NTU(1−Cr) (Cr<1);εpf=1−e−NTU(1+Cr)1+Cr\varepsilon_{cf}=\frac{1-e^{-NTU(1-C_r)}}{1-C_r e^{-NTU(1-C_r)}}\ (C_r<1);\quad\varepsilon_{pf}=\frac{1-e^{-NTU(1+C_r)}}{1+C_r}Th,o=Th,i−Q˙/Ch,Tc,o=Tc,i+Q˙/CcT_{h,o}=T_{h,i}-\dot Q/C_h,\quad T_{c,o}=T_{c,i}+\dot Q/C_c
Keep the hardware; change the flow arrangementConnect the equipment to the calculation

01 · Equipment view

A realistic double-pipe exchanger cutaway reveals separate inner-tube and annular flow passages and four fluid connections.

Hot stream · cold stream · two separated passages

The cutaway shows separate flow passages. The original schematic defines the inlet states and arrangement for each teaching model.

02 · Exam-style model

Zoom in to read every label.

Keep the hardware; change the flow arrangement

100%Open in a new tab

Scroll or swipe inside the viewport. Read the units and assumptions with the labelled components.

Two original exchanger schematics compare opposite-end counterflow inlets with same-end parallel inlets. Both have 160°F hot and 60°F cold inlets, identical capacity rates and UA, and unknown outlets.

Both cases share the inlet temperatures, capacity rates and UA. Opposed arrows define counterflow; aligned arrows define parallel flow. Outlet temperatures remain unknown until the model is solved.

Original teaching figure. The schematic and stated givens define the model; the 3D view illustrates the hardware. These are not actual examination exhibits.

Study calculator, scratch notepad and whiteboard

Supports +, −, *, /, ^, parentheses, sqrt, abs, log (base 10), ln, sin, cos, tan, pi and e. Confirm units independently.

Scratch text and drawings stay in this tab and clear when you leave or reload. Save important reasoning in a response or reflection below before leaving.

Scratch whiteboard

Sketch a control volume or air path with a mouse, pen or touch. Ink stays in this page session and clears on reload. Use the response and reflection fields as a keyboard alternative. Sign in before starting and save a reflection to retain your reasoning.

1 · Choose the method

For the specified counterflow exchanger with both outlets unknown, which route determines duty without inventing terminal temperatures?

Choose one response

2 · Change the condition

Predict before you inspect the calculation.

Change the arrangement to parallel flow; preserve both inlet states, capacity rates and UA.

  • Parallel flow; hot inlet 160°F, cold inlet 60°F, ChC_{h} = 2,400 and CcC_{c} = 1,200 Btu/(h·°F), UA = 1,800 Btu/(h·°F).
  • Use the supplied parallel-flow expression; constant properties and no loss.

Check the first decision to open the changed-condition response.

Personal notes are not automatically graded. Workshops are linked learning experiments, separate from timed diagnostics and mocks. They do not predict a pass or award instructional hours.

Reference navigation drill

Search concept: effectiveness, number of transfer units

Locate the arrangement-specific relationship in your authorized reference. Identify which stream supplies C_min and how the equal-capacity limit is handled.

Use an authorized reference. A search term does not guarantee that a current examination reference contains every relationship used in this teaching case.

Where this model stops

Original idealized sensible models. Shell passes, crossflow mixing, fouling, phase change and variable properties require the appropriate additional model. The schematic supplies givens, not calculated outlets.

The PE Mechanical: HVAC and Refrigeration examination uses the NCEES PE Mechanical Reference Handbook. These source links provide study context; they are not additional exam-day references.

Found an unclear condition or calculation? Include workshop exchanger-arrangement and the step in your comment.

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