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Heat Exchanger Analysis11:38

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Source: Alexander S Rattner and Christopher J Greer; Department of Mechanical and Nuclear Engineering, The Pennsylvania State University, University Park, PA
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Derivatives quantify the rate of change of a function and can be interpreted geometrically as the slope of a straight line or the slope of a tangent line to a curve at a given point. In the context of a roller coaster, the derivative of the function describing the track’s horizontal position provides a mathematical description of how steep the path is at any location along the ride.Constant and Linear PathsA horizontal segment of a roller coaster can be modeled by a constant function,...
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The range is one of the measures of variation. It can be defined as the difference between a dataset's highest and lowest values. For example, in the study of seven 16-ounce soda cans, the filled volume of soda was measured, thus producing the following amount (in ounces) of soda:
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To achieve precise distance measurements, especially in surveying and construction, certain corrections must be applied to account for potential sources of error like the standardization errors, temperature variations, and slope adjustments.Standardization error emerges when measurement equipment undergoes changes, such as wear, repairs, or weather impacts. To address this, surveyors compare the equipment’s readings to a standard. This process identifies any deviation that might lead to...
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Related Experiment Video

Updated: Jan 20, 2026

Rating and Modeling Procedures for Heat Exchangers
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Rating and Modeling Procedures for Heat Exchangers

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Simple exchange hole models for long-range-corrected density functionals.

Dimitri N Laikov1

  • 1Chemistry Department, Moscow State University, 119991 Moscow, Russia.

The Journal of Chemical Physics
|September 8, 2019
PubMed
Summary
This summary is machine-generated.

New density functional models improve short-range exchange calculations by interpolating between uniform electron gas and Hermite function limits. These models offer explicit formulas for improved accuracy in generalized gradient approximations.

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Area of Science:

  • Computational chemistry
  • Quantum chemistry
  • Density functional theory

Background:

  • Range-separated density functionals offer improvements over semilocal and hybrid functionals.
  • Converting semilocal functionals to short-range analogs is complex due to increased information content.

Purpose of the Study:

  • Develop simple models for the spherically averaged exchange hole.
  • Create explicit, closed-form expressions for energy density using elementary and error functions.
  • Evaluate new models against existing ones and exact limits.

Main Methods:

  • Interpolation between uniform electron gas and few-term Hermite function models for the exchange hole.
  • Development of generalized gradient approximations (GGAs) incorporating these models.
  • Derivation of explicit formulas for the error-function-weighted Coulomb interaction energy density.
  • Comparison with non-oscillatory models and the exact uniform electron gas limit.

Main Results:

  • New models provide explicit closed-form expressions for energy density.
  • Energy densities from developed models closely match (within 5%) non-oscillatory models.
  • The developed models show a slight mismatch (approx. 1%) with the exact uniform electron gas limit.

Conclusions:

  • The developed models offer a practical approach to implementing short-range exchange in density functional theory.
  • These models provide a balance between accuracy and computational tractability for GGAs.
  • Further refinement may be needed to perfectly match the uniform electron gas limit.