This file contains notes on the important topic of EES Functions for Heat Exchanger calculations. Some solved problems are also included.
These notes were prepared while teaching Heat Transfer course to the M.Tech. students in Mechanical Engineering Dept. of St. Joseph Engineering College, Vamanjoor, Mangalore, India.
It is hoped that these notes will be useful to teachers, students, researchers and professionals working in this field.
Contents:
• Overall heat transfer coefficient
• Importance of ‘Fouling factor’
• Analysis of heat exchangers by ‘Logarithmic Mean Temp Difference (LMTD)’ method
• Correction factors for Cross-flow and Shell & Tube heat exchangers
• Analysis of heat exchangers by ‘No. of Transfer Units (NTU)– Effectiveness (ε)’ method
• Compact heat exchangers
EES Functions and Procedures for Forced convection heat transfertmuliya
This file contains notes on Engineering Equation Solver (EES) Functions and Procedures for Forced convection heat transfer calculations. Some problems are also included.
These notes were prepared while teaching Heat Transfer course to the M.Tech. students in Mechanical Engineering Dept. of St. Joseph Engineering College, Vamanjoor, Mangalore, India.
It is hoped that these notes will be useful to teachers, students, researchers and professionals working in this field.
Contents:
• Forced convection – Tables of formulas
• Boundary layer, flow over flat plates, across cylinders, spheres and tube banks –
• Flow inside tubes and ducts
Boiling and Condensation heat transfer -- EES Functions and Procedurestmuliya
This file contains notes on Engineering Equation Solver (EES) Functions and Procedures for Boiling and Condensation heat transfer. Some problems are also included.
These notes were prepared while teaching Heat Transfer course to the M.Tech. students in Mechanical Engineering Dept. of St. Joseph Engineering College, Vamanjoor, Mangalore, India.
Contents: Summary of formulas used -
EES Functions/Procedures for boiling: Nucleate boiling heat flux for any geometry - critical heat flux for large horizontal surface, horizontal cylinder and sphere - Film boiling for horizontal cylinder, sphere and horizontal surface – Problems.
EES Functions/Procedures for condensation of: steam on vertical surface – any fluid on a vertical surface – steam on vertical cylinder – any fluid on vertical cylinder – steam on horizontal cylinder – any fluid on horizontal cylinder – steam on a horizontal tube bank – any fluid on horizontal tube bank – any fluid on a sphere – any fluid inside a horizontal cylinder - Problems.
It is hoped that these notes will be useful to teachers, students, researchers and professionals working in this field.
EES Functions and Procedures for Natural convection heat transfertmuliya
This file contains notes on Engineering Equation Solver (EES) Functions and Procedures for Natural (or, free) convection heat transfer calculations. Some problems are also included.
These notes were prepared while teaching Heat Transfer course to the M.Tech. students in Mechanical Engineering Dept. of St. Joseph Engineering College, Vamanjoor, Mangalore, India.
It is hoped that these notes will be useful to teachers, students, researchers and professionals working in this field.
Contents:
• Natural convection formulas - Tables
• Natural convection from Vertical plates & cylinders, horizontal plates, cylinders and spheres, from enclosed spaces, rotating disks and spheres, and from finned surfaces
• Combined Natural and forced convection
Performing Manual and Automated Iterations in Engineering Equation Solver (EES) - Examples from Heat Transfer.
All the EES codes shown in the examples are available at: https://goo.gl/KExGFi
EES Functions and Procedures for Forced convection heat transfertmuliya
This file contains notes on Engineering Equation Solver (EES) Functions and Procedures for Forced convection heat transfer calculations. Some problems are also included.
These notes were prepared while teaching Heat Transfer course to the M.Tech. students in Mechanical Engineering Dept. of St. Joseph Engineering College, Vamanjoor, Mangalore, India.
It is hoped that these notes will be useful to teachers, students, researchers and professionals working in this field.
Contents:
• Forced convection – Tables of formulas
• Boundary layer, flow over flat plates, across cylinders, spheres and tube banks –
• Flow inside tubes and ducts
Boiling and Condensation heat transfer -- EES Functions and Procedurestmuliya
This file contains notes on Engineering Equation Solver (EES) Functions and Procedures for Boiling and Condensation heat transfer. Some problems are also included.
These notes were prepared while teaching Heat Transfer course to the M.Tech. students in Mechanical Engineering Dept. of St. Joseph Engineering College, Vamanjoor, Mangalore, India.
Contents: Summary of formulas used -
EES Functions/Procedures for boiling: Nucleate boiling heat flux for any geometry - critical heat flux for large horizontal surface, horizontal cylinder and sphere - Film boiling for horizontal cylinder, sphere and horizontal surface – Problems.
EES Functions/Procedures for condensation of: steam on vertical surface – any fluid on a vertical surface – steam on vertical cylinder – any fluid on vertical cylinder – steam on horizontal cylinder – any fluid on horizontal cylinder – steam on a horizontal tube bank – any fluid on horizontal tube bank – any fluid on a sphere – any fluid inside a horizontal cylinder - Problems.
It is hoped that these notes will be useful to teachers, students, researchers and professionals working in this field.
EES Functions and Procedures for Natural convection heat transfertmuliya
This file contains notes on Engineering Equation Solver (EES) Functions and Procedures for Natural (or, free) convection heat transfer calculations. Some problems are also included.
These notes were prepared while teaching Heat Transfer course to the M.Tech. students in Mechanical Engineering Dept. of St. Joseph Engineering College, Vamanjoor, Mangalore, India.
It is hoped that these notes will be useful to teachers, students, researchers and professionals working in this field.
Contents:
• Natural convection formulas - Tables
• Natural convection from Vertical plates & cylinders, horizontal plates, cylinders and spheres, from enclosed spaces, rotating disks and spheres, and from finned surfaces
• Combined Natural and forced convection
Performing Manual and Automated Iterations in Engineering Equation Solver (EES) - Examples from Heat Transfer.
All the EES codes shown in the examples are available at: https://goo.gl/KExGFi
This chapter contains:-.
Analytical Methods of two dimensional steady state heat conduction
Finite difference Method application on two dimensional steady state heat conduction.
Finite difference method on irregular shape of a system
Heat transfer from extended surfaces (or fins)tmuliya
This file contains slides on Heat Transfer from Extended Surfaces (FINS). The slides were prepared while teaching Heat Transfer course to the M.Tech. students in Mechanical Engineering Dept. of St. Joseph Engineering College, Vamanjoor, Mangalore, India.
Contents: Governing differential eqn – different boundary conditions – temp. distribution and heat transfer rate for: infinitely long fin, fin with insulated end, fin losing heat from its end, and fin with specified temperatures at its ends – performance of fins - ‘fin efficiency’ and ‘fin effectiveness’ – fins of non-uniform cross-section- thermal resistance and total surface efficiency of fins – estimation of error in temperature measurement - Problems
In this Thesis I will try to understand the concept associated with cooling towers and model a laboratory sized cooling tower in a software package called Engineering Equation Solver (EES). An example of system modelling is presented in this progress report, along with the comparison of a set of results with an experimental data from P.A Hilton Model H892 Bench top cooling tower with a maximum of 9% error. A user interface is also modelled to simulate off-design performance rather than conducting experiments. It also allows you to do additional scenarios that cannot be practically being done in lab,
like Relative humidity, etc.
Aircraft refrigeration system (air cooling system)Ripuranjan Singh
Aircraft air refrigeration systems are required due to heat transfer from many external and internal heat sources (like solar radiation and avionics) which increase the cabin air temperature. With the technological developments in high-speed passenger and jet aircraft's, the air refrigeration systems are proving to be most efficient, compact and simple. Various types of aircraft air refrigeration systems used these days are.
Simple air cooling system
Simple air evaporative cooling system
Boot strap air cooling system
Boot strap air evaporative cooling system
Reduced ambient air cooling system
Regenerative air cooling system
COMPRESSOR EFFICIENCY AND TURBINE EFFICIENCY.
Comparison of Various Air Cooling Systems used for Aircraft ON basis of dart
PPTs deals with the Unit 5 of Power Plant Engineering, discusses Load Curve, Load duration curve, various factors associated with power palnt like Load factor, capacity factor, use factor, demand factor , diversity factor, method of calculating different costs involved in power generation, differential fuel costing and its implications on sharing of units, factors determine the selection of site for Power plants Workedout problems are also dealt
A review of 1st and 2nd Law of Thermodynamics is presented, as well as the introduction to the concepts of availability and exergy. Examples of calculations for power plants are presented using both analytical methods and CyclePad(TM) simulator.
Characteristics of single pump and pumps in series and parallel use of indust...TOPENGINEERINGSOLUTIONS
This is a water engineering assignment on Characteristics of single pump and pumps in series and parallel
(Use of Industry Standard Software)
Module Code: NG2S106, Module Title: Water Engineering
This chapter contains:-.
Analytical Methods of two dimensional steady state heat conduction
Finite difference Method application on two dimensional steady state heat conduction.
Finite difference method on irregular shape of a system
Heat transfer from extended surfaces (or fins)tmuliya
This file contains slides on Heat Transfer from Extended Surfaces (FINS). The slides were prepared while teaching Heat Transfer course to the M.Tech. students in Mechanical Engineering Dept. of St. Joseph Engineering College, Vamanjoor, Mangalore, India.
Contents: Governing differential eqn – different boundary conditions – temp. distribution and heat transfer rate for: infinitely long fin, fin with insulated end, fin losing heat from its end, and fin with specified temperatures at its ends – performance of fins - ‘fin efficiency’ and ‘fin effectiveness’ – fins of non-uniform cross-section- thermal resistance and total surface efficiency of fins – estimation of error in temperature measurement - Problems
In this Thesis I will try to understand the concept associated with cooling towers and model a laboratory sized cooling tower in a software package called Engineering Equation Solver (EES). An example of system modelling is presented in this progress report, along with the comparison of a set of results with an experimental data from P.A Hilton Model H892 Bench top cooling tower with a maximum of 9% error. A user interface is also modelled to simulate off-design performance rather than conducting experiments. It also allows you to do additional scenarios that cannot be practically being done in lab,
like Relative humidity, etc.
Aircraft refrigeration system (air cooling system)Ripuranjan Singh
Aircraft air refrigeration systems are required due to heat transfer from many external and internal heat sources (like solar radiation and avionics) which increase the cabin air temperature. With the technological developments in high-speed passenger and jet aircraft's, the air refrigeration systems are proving to be most efficient, compact and simple. Various types of aircraft air refrigeration systems used these days are.
Simple air cooling system
Simple air evaporative cooling system
Boot strap air cooling system
Boot strap air evaporative cooling system
Reduced ambient air cooling system
Regenerative air cooling system
COMPRESSOR EFFICIENCY AND TURBINE EFFICIENCY.
Comparison of Various Air Cooling Systems used for Aircraft ON basis of dart
PPTs deals with the Unit 5 of Power Plant Engineering, discusses Load Curve, Load duration curve, various factors associated with power palnt like Load factor, capacity factor, use factor, demand factor , diversity factor, method of calculating different costs involved in power generation, differential fuel costing and its implications on sharing of units, factors determine the selection of site for Power plants Workedout problems are also dealt
A review of 1st and 2nd Law of Thermodynamics is presented, as well as the introduction to the concepts of availability and exergy. Examples of calculations for power plants are presented using both analytical methods and CyclePad(TM) simulator.
Characteristics of single pump and pumps in series and parallel use of indust...TOPENGINEERINGSOLUTIONS
This is a water engineering assignment on Characteristics of single pump and pumps in series and parallel
(Use of Industry Standard Software)
Module Code: NG2S106, Module Title: Water Engineering
Mathcad Functions for Forced convection heat transfer calculationstmuliya
This file contains notes on Mathcad Functions for Forced convection heat transfer calculations. Some problems are also included.
These notes were prepared while teaching Heat Transfer course to the M.Tech. students in Mechanical Engineering Dept. of St. Joseph Engineering College, Vamanjoor, Mangalore, India. It is hoped that these notes will be useful to teachers, students, researchers and professionals working in this field.
Contents: Forced convection formulas – boundary layer, flow over flat plates, across cylinders, spheres and tube banks
Lectures on Heat Transfer - Introduction - Applications - Fundamentals - Gove...tmuliya
This file contains Introduction to Heat Transfer and Fundamental laws governing heat transfer.
The slides were prepared while teaching Heat Transfer course to the M.Tech. students in Mechanical Engineering Dept. of St. Joseph Engineering College, Vamanjoor, Mangalore, India.
Hydrologic analysis and design 4th edition mc cuen solutions manualMacionis99
Hydrologic Analysis and Design 4th Edition McCuen Solutions Manual
Full download: https://goo.gl/2LAVj2
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Mathcad Functions for Conduction heat transfer calculationstmuliya
This file contains notes on Mathcad Functions for Conduction heat transfer calculations. Some problems are also included.
These notes were prepared while teaching Heat Transfer course to the M.Tech. students in Mechanical Engineering Dept. of St. Joseph Engineering College, Vamanjoor, Mangalore, India.
It is hoped that these notes will be useful to teachers, students, researchers and professionals working in this field.
Mathcad Functions for Natural (or free) convection heat transfer calculationstmuliya
This file contains notes on Mathcad Functions for Natural (or, free) convection heat transfer calculations. Some problems are also included.
These notes were prepared while teaching Heat Transfer course to the M.Tech. students in Mechanical Engineering Dept. of St. Joseph Engineering College, Vamanjoor, Mangalore, India.
It is hoped that these notes will be useful to teachers, students, researchers and professionals working in this field.
Contents: Free convection from vertical plates and cylinders, horizontal plates, cylinders and spheres, enclosed spaces, rotating cylinders, disks and spheres. Finned surfaces.
Combined Natural and Forced convection.
Mathcad functions for fluid properties - for convection heat transfer calcu...tmuliya
This file contains slides on Mathcad Functions for Fluid properties--- Useful in Convection heat transfer calculations.
The slides were prepared while teaching Heat Transfer course to the M.Tech. students in Mechanical Engineering Dept. of St. Joseph Engineering College, Vamanjoor, Mangalore, India, during Sept. – Dec. 2010.
It is hoped that these Slides will be useful to teachers, students, researchers and professionals working in this field.
Contents: Mathcad Functions for properties of:
Air, Ammonia, Ethyl alcohol, Ethylene glycol, Un-used engine oil, Glycerin, Lead, and Mercury
Mathcad Functions for Boiling heat transfertmuliya
This file contains slides on Mathcad Functions for Boiling heat transfer.
The slides were prepared while teaching Heat Transfer course to the M.Tech. students in Mechanical Engineering Dept. of St. Joseph Engineering College, Vamanjoor, Mangalore, India, during Sept. – Dec. 2010.
Contents: Functions for properties of sat. water and steam- Nucleate boiling for water – heat flux – crit. heat flux for: flat surface, horizl. cylinder – min. heat flux for horizl. surface – Film boiling for horizl. surface , very large dia tube, horizl. cylinder, sphere - Nucleate boiling of water for submerged horizl. and vertical surfaces
Mathcad Functions for Condensation heat transfertmuliya
This file contains slides on Mathcad Functions for Condensation heat transfer.
The slides were prepared while teaching Heat Transfer course to the M.Tech. students in Mechanical Engineering Dept. of St. Joseph Engineering College, Vamanjoor, Mangalore, India, during Sept. – Dec. 2010.
Contents: Functions for properties of sat. water and steam- Film condensation of steam on vertical plate and inclined plate – on vertical cylinder and horizontal cylinder – on horizontal cylinders in vertical tier - on a sphere – inside horizontal tubes – on copper surfaces – Mathcad Functions for properties of sat. Ammonia – Film condensation of Ammonia on vertical plate, horizontal cylinder and tube banks, and inside horizontal tubes
Thermal Radiation - III- Radn. energy exchange between gray surfacestmuliya
This file contains slides on THERMAL RADIATION-III: Radiation energy exchange between gray surfaces.
The slides were prepared while teaching Heat Transfer course to the M.Tech. students in Mechanical Engineering Dept. of St. Joseph Engineering College, Vamanjoor, Mangalore, India, during Sept. – Dec. 2010.
Contents: Radiation heat exchange between gray surfaces - electrical network method – two zone enclosures – Problems - three zone enclosures – Problems - radiation shielding – Problems - radiation error in temperature measurement – Problems
Thermal Radiation-II- View factors and Radiation energy exchange between blac...tmuliya
This file contains slides on THERMAL RADIATION-II: View factors and Radiation energy exchange between black bodies.
The slides were prepared while teaching Heat Transfer course to the M.Tech. students in Mechanical Engineering Dept. of St. Joseph Engineering College, Vamanjoor, Mangalore, India, during Sept. – Dec. 2010.
Contents: View factor – general relations – radiation energy exchange between black bodies – properties of view factor and view factor algebra – view factor formulas and graphs – Problems
Thermal Radiation-I - Basic properties and Lawstmuliya
This file contains slides on RADIATION-I: Basic Properties and Laws.
The slides were prepared while teaching Heat Transfer course to the M.Tech. students in Mechanical Engineering Dept. of St. Joseph Engineering College, Vamanjoor, Mangalore, India, during Sept. – Dec. 2010.
Contents: Introduction – Applications – Electromagnetic spectrum – Properties and definitions – Laws of black body radiation – Planck’s Law – Wein’s displacement law – Stefan Boltzmann Law – Radiation from a wave band – Emissivity – Kirchoff’s Law- Problems
This file contains slides on Steady State Heat Conduction in Multiple Dimensions.
The slides were prepared while teaching Heat Transfer course to the M.Tech. students in Mechanical Engineering Dept. of St. Joseph Engineering College, Vamanjoor, Mangalore, India, during Sept. – Dec. 2010.
Contents: 2-D conduction - Various methods of solution – Analytical - Graphical - Analogical – Numerical – Shape factors for 2-D conduction - Problems
Numerical methods in Transient-heat-conductiontmuliya
This file contains slides on Numerical methods in Transient heat conduction.
The slides were prepared while teaching Heat Transfer course to the M.Tech. students in Mechanical Engineering Dept. of St. Joseph Engineering College, Vamanjoor, Mangalore, India, during Sept. – Dec. 2010.
Contents: Finite difference eqns. by energy balance – Explicit and Implicit methods – 1-D transient conduction in a plane wall – stability criterion – Problems - 2-D transient heat conduction – Finite diff. eqns. for interior nodes – Explicit and Implicit methods - stability criterion – difference eqns for different boundary conditions – Accuracy considerations – discretization error and round–off error - Problems
NUMERICAL METHODS IN STEADY STATE, 1D and 2D HEAT CONDUCTION- Part-IItmuliya
This file contains slides on NUMERICAL METHODS IN STEADY STATE 1D and 2D HEAT CONDUCTION – Part-II.
The slides were prepared while teaching Heat Transfer course to the M.Tech. students in Mechanical Engineering Dept. of St. Joseph Engineering College, Vamanjoor, Mangalore, India, during Sept. – Dec. 2010.
Contents: Methods of solving a system of simultaneous, algebraic equations - 1D steady state conduction in cylindrical and spherical systems - 2D steady state conduction in cartesian coordinates - Problems
This file contains slides on NUMERICAL METHODS IN STEADY STATE 1D and 2D HEAT CONDUCTION – Part-I.
The slides were prepared while teaching Heat Transfer course to the M.Tech. students.
Contents: Why Numerical methods? – Advantages – Finite difference formulation from differential eqns – 1D steady state conduction in cartesian coordinates – formulation by energy balance method – different BC’s – Problems
This file contains slides on Transient Heat conduction: Part-II
The slides were prepared while teaching Heat Transfer course to the M.Tech. students in the year 2010.
Contents: Semi-infinite solids with different BC’s - Problems - Product solution for multi-dimension systems -
Summary of Basic relations for transient conduction
This file contains slides on Transient Heat conduction: Part-I
The slides were prepared while teaching Heat Transfer course to the M.Tech. students in Mechanical Engineering Dept. of St. Joseph Engineering College, Vamanjoor, Mangalore, India, during Sept. – Dec. 2010. Contents: Lumped system analysis – criteria for lumped system analysis – Biot and Fourier Numbers – Response time of a thermocouple - One-dimensional transient conduction in large plane walls, long cylinders and spheres when Bi > 0.1 – one-term approximation - Heisler and Grober charts- Problems
One dim, steady-state, heat conduction_with_heat_generationtmuliya
This file contains slides on One-dimensional, steady-state heat conduction with heat generation.
The slides were prepared while teaching Heat Transfer course to the M.Tech. students in Mechanical Engineering Dept. of St. Joseph Engineering College, Vamanjoor, Mangalore, India, during Sept. – Dec. 2010.
It is hoped that these Slides will be useful to teachers, students, researchers and professionals working in this field.
This file contains slides on One-dimensional, steady state heat conduction without heat generation. The slides were prepared while teaching Heat Transfer course to the M.Tech. students.
Topics covered: Plane slab - composite slabs – contact resistance – cylindrical Systems – composite cylinders - spherical systems – composite spheres - critical thickness of insulation – optimum thickness – systems with variable thermal conductivity
Industrial Training at Shahjalal Fertilizer Company Limited (SFCL)MdTanvirMahtab2
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This paper addresses the vulnerability of deep learning models, particularly convolutional neural networks
(CNN)s, to adversarial attacks and presents a proactive training technique designed to counter them. We
introduce a novel volumization algorithm, which transforms 2D images into 3D volumetric representations.
When combined with 3D convolution and deep curriculum learning optimization (CLO), itsignificantly improves
the immunity of models against localized universal attacks by up to 40%. We evaluate our proposed approach
using contemporary CNN architectures and the modified Canadian Institute for Advanced Research (CIFAR-10
and CIFAR-100) and ImageNet Large Scale Visual Recognition Challenge (ILSVRC12) datasets, showcasing
accuracy improvements over previous techniques. The results indicate that the combination of the volumetric
input and curriculum learning holds significant promise for mitigating adversarial attacks without necessitating
adversary training.
Hierarchical Digital Twin of a Naval Power SystemKerry Sado
A hierarchical digital twin of a Naval DC power system has been developed and experimentally verified. Similar to other state-of-the-art digital twins, this technology creates a digital replica of the physical system executed in real-time or faster, which can modify hardware controls. However, its advantage stems from distributing computational efforts by utilizing a hierarchical structure composed of lower-level digital twin blocks and a higher-level system digital twin. Each digital twin block is associated with a physical subsystem of the hardware and communicates with a singular system digital twin, which creates a system-level response. By extracting information from each level of the hierarchy, power system controls of the hardware were reconfigured autonomously. This hierarchical digital twin development offers several advantages over other digital twins, particularly in the field of naval power systems. The hierarchical structure allows for greater computational efficiency and scalability while the ability to autonomously reconfigure hardware controls offers increased flexibility and responsiveness. The hierarchical decomposition and models utilized were well aligned with the physical twin, as indicated by the maximum deviations between the developed digital twin hierarchy and the hardware.
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
Explore the innovative world of trenchless pipe repair with our comprehensive guide, "The Benefits and Techniques of Trenchless Pipe Repair." This document delves into the modern methods of repairing underground pipes without the need for extensive excavation, highlighting the numerous advantages and the latest techniques used in the industry.
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$UnitSystem SI Pa C J
PROCEDURE Overall_U_PlaneWall ( L, k_wall,A,h_i,h_0,R_fi,R_f0: R_cond, R_i, R_0, R_ci, R_c0,
R_tot,U)
"Ref: Cengel"
"Finds U and various thermal resistances for a Plane Wall:"
"Inputs: L (m), k_wall (W/m.C), A (m^2),h_i, h_0 (W/m2.C), fouling factors: R_fi, R_f0 (m2.C/W)"
"Outputs: R_cond,R_i, R_0, R_ci, R_c0, R_tot (C/W), U (W/m2.C)"
R_cond:= L/(k_wall * A) "condn. resist."
R_i:= 1/(h_i * A) "inside conv. resist."
R_0:= 1/(h_0 * A) "outside conv. resist."
R_ci := R_fi/A "inside fouling resist."
R_c0:= R_f0/A "outside fouling resist."
R_tot:= R_cond + R_i + R_ci + R_c0 + R_0 "total thermal resist."
U:= 1/(A * R_tot) "Overall heat transfer coeff."
END
"======================================================"
6. $UnitSystem SI Pa C J
PROCEDURE Overall_U_cyl ( L, k_wall, D_i, D_0, h_i, h_0, R_fi, R_f0: R_cond, R_i, R_0, R_ci, R_c0,
R_tot, U_i, U_0)
"Ref: Cengel"
"Finds U_i and U_0, and various thermal resistances for a cylindrical Wall:"
"Inputs: L (m), k_wall (W/m.C), D_i, D_0 (m),h_i, h_0 (W/m2.C), fouling factors: R_fi, R_f0 (m2.C/W)"
"Outputs: R_cond,R_i, R_0, R_ci, R_c0, R_tot (C/W), U_i, U_0 (W/m2.C)"
A_i:= pi * D_i * L "m^2, inside surface area"
A_0:= pi * D_0 * L "m^2, outside surface area"
R_cond:= ln(D_0/D_i) / (2 * pi * k_wall * L) "condn. resist."
R_i:= 1/(h_i * A_i) "inside conv. resist."
R_0:= 1/(h_0 * A_0) "outside conv. resist."
R_ci := R_fi/A_i "inside fouling resist."
R_c0:= R_f0/A_0 "outside fouling resist."
R_tot:= R_cond + R_i + R_ci + R_c0 + R_0 "total thermal resist."
U_i:= 1/(A_i * R_tot) "Overall heat transfer coeff based on inner heat tr area."
U_0:= 1/(A_0 * R_tot) "Overall heat transfer coeff based on outer heat tr area."
END
"======================================================"
$UnitSystem SI Pa C J
PROCEDURE ForcedConv_AcrossCylinder (Fluid$,P_infinity, T_infinity, U_infinity, L, D, T_s: Re_D,
Nusselt_D_bar, h_bar, Q)
"Ref: Incropera, 5th Ed. pp. 411, Eqn. (7.57)"
"Churchill and Bernstein eqn....for entire range of Re_D and a wide range of Pr"
"Finds various quantities for flow of Air or any Fluid across a cylinder:"
"Inputs: Pa, C, m/s, m"
"Outputs: W/m^2.C, W, W"
T_f := (T_infinity + T_s)/2 " mean film temp, C"
7. "Properties of Air (Ideal gas) or other Fluid at T_f :"
IF Fluid$ = 'Air' Then
rho:=Density(Fluid$,T=T_f,P=P_infinity)
mu:=Viscosity(Fluid$,T=T_f)
k:=Conductivity(Fluid$,T=T_f)
Pr:=Prandtl(Fluid$,T=T_f)
cp:=SpecHeat(Fluid$,T=T_f)
ELSE
rho:=Density(Fluid$,T=T_f,P=P_infinity)
mu:=Viscosity(Fluid$,T=T_f,P=P_infinity)
k:=Conductivity(Fluid$,T=T_f,P=P_infinity)
Pr:=Prandtl(Fluid$,T=T_f,P=P_infinity)
cp:=SpecHeat(Fluid$,T=T_f,P=P_infinity)
ENDIF
Re_D := D * U_infinity * rho/mu "Finds Reynolds No."
"To find h accurately: Use Churchill and Bernstein eqn."
Nusselt_D_bar := 0.3 + ((0.62 * Re_D^0.5 * (Pr)^(1/3))/(1 + (0.4/Pr)^(2/3))^(1/4)) * (1 +
(Re_D/282000)^(5/8))^(4/5)
h_bar :=Nusselt_D_bar * k / D "Finds h_bar"
Q := h_bar * (pi * D * L) * (T_s - T_infinity) "W.... heat tr"
END
"======================================================================"
"Prob. B.1.1: A shell and tube counter-flow heat exchanger uses copper tubes (k = 380 W/(m.C)), 20 mm
ID and 23 mm OD. Inside and outside film coefficients are 5000 and 1500 W/(m2.C) respectively.
Fouling factors on the inside and outside may be taken as 0.0004 and 0.001 m2.C/W respectively.
Calculate the overall heat transfer coefficient based on: (i) outside surface, and (ii) inside surface."
"Data:"
D_i = 0.02 [m]
D_0 = 0.023 [m]
L = 1 [m]
k_wall = 380 [W/m_C]
h_i = 5000 [W/m^2-C] "...heat tr coeff on the inside"
h_0 = 1500 [W/m^2-C] "...heat tr coeff on the outside"
R_fi = 0.0004 [m^2-C/W] "...Fouling factor on the inside"
11. "Prob. B.1.2: Consider a type 302 SS tube (k = 15.10 W/(m.C)), 22 mm ID and 27 mm OD, inside
which water flows at a mean temp T_m = 75 C and velocity u_m = 0.5 m/s. Air at 15 C and at a velocity
of V_0 = 20 m/s flows across this tube. Fouling factors on the inside and outside may be taken as 0.0004
and 0.0002 m^2.C/W respectively. Determine the overall heat transfer coefficient based on the outside
surface, U_0 (b) Plot U_o as a function of cross flow velocity, V_0 in the range: 5 < V_o < 30 m/s."
"EES Solution:"
"Data:"
D_i = 0.022 [m]
D_0 = 0.027 [m]
L = 1 [m]
k_wall = 15.1 [W/m-C]
T_m_water = 75 [C]
U_m_water = 0.5 [m/s]
P_1 = 1.01325E05 [Pa]
T_m_air = 15 [C]
V_0_air = 20 [m/s]
"Properties of Water at T_m:"
rho_w=Density(Steam_IAPWS,T=T_m_water,P=P_1)
mu_w=Viscosity(Steam_IAPWS,T=T_m_water,P=P_1)
cp_w=SpecHeat(Steam_IAPWS,T=T_m_water,P=P_1)
k_w=Conductivity(Steam_IAPWS,T=T_m_water,P=P_1)
Pr_w=Prandtl(Steam_IAPWS,T=T_m_water,P=P_1)
"Calculations:"
"To determine inside heat transfer coeff. h_i:"
Re_water = D_i * U_m_water * rho_w / mu_w "...finds Reynolds No. for water"
"Re_water = 28388 > 4000; So, apply Dittus - Boelter eqn to find out Nusselts No.:"
Nusselts_w = 0.023 * Re_water^0.8 * Pr_w^0.4 "...gives Nusselts No."
Nusselts_w = h_i * D_i /k_w "....finds h_i, heat tr coeff on the inside "
"To determine outside heat transfer coeff. h_0:"
"It is cross flow of air across a cylinder. So, use the EES PROCEDURE written above to find out h_0,
using the Churchill - Bernstein eqn for cross flow of a fluid over a cylinder:"
12. Fluid$ = 'Air'
P_infinity = P_1
T_infinity = 15[C]
U_infinity = V_0_air
D = D_0
{T_s = 70 "[C] .... assumed, will be corrected later"}
CALL ForcedConv_AcrossCylinder (Fluid$,P_infinity, T_infinity, U_infinity, L, D, T_s: Re_D,
Nusselt_D_bar, h_0, Q)
R_fi = 0.0004 [m^2-C/W] "...Fouling factor on the inside"
R_f0 = 0.0002 [m^2-C/W] "...Fouling factor on the outside"
CALL Overall_U_cyl ( L, k_wall, D_i, D_0, h_i, h_0, R_fi, R_f0: R_cond, R_i, R_0, R_ci, R_c0, R_tot, U_i,
U_0)
{(T_s - T_m_air) / R_conv_out = (T_m_water - T_s) / (R_conv_in + R_c_in + R_cond_wall + R_c_out)
"....finds T_s, the surface temp of cylinder"}
(T_s - 15[C]) / R_0 = (T_m_water - T_s) / (R_i +R_ci + R_cond + R_c0) "Finds surface temp on the
fouling layer on the outer surface of the tube "
U_i * A_i = 1 / R_tot "...determine U _i"
U_0 * A_0 = 1 / R_tot "...determine U _0"
Results:
14. 3 & ; # # , ; 5
"Prob. B.2.1. A HX is required to cool 55000 kg/h of alcohol from 66 C to 40 C using 40000 kg/h of
water entering at 5 C. Calculate the following: (i) the exit temp of water (ii) surface area required for
parallel flow and counter-flow HXs. Take U = 580 W/m^2.K, cp for alcohol = 3760 J/kg.K, cp for water
= 4180 J/kg.K. [VTU -May - June- 2006]"
Fig. Prob.B.2.1(a).Counter-flow arrangement
Fig. Prob.B.2.1(b).Parallel flow arrangement
EES Solution:
"Data:"
m_h = 55000 [kg/h] * convert (kg/h, kg/s) "...hot fluid---alcohol"
m_c = 40000 [kg/h] * convert (kg/h, kg/s) "....cold fluid -- water"
T_h_i = 66 [C]
T_c_i = 5 [C]
T_h_o = 40 [C]
cp_h = 3760 [J/kg-C]
15. cp_c = 4180 [J/kg-C]
U= 580 [W/m^2-C]
"Calculations:"
"Exit temp of cold fluid:"
m_h * cp_h * (T_h_i - T_h_o) = m_c * cp_c * (T_c_o - T_c_i) " C...determines T_c_o"
"LMTD for a counter-flow HX:"
DELTAT_1 = T_h_i - T_c_o " Temp diff at inlet of HX --- for counter flow HX"
DELTAT_2 = T_h_o - T_c_i " Temp diff at exit of HX --- for counter flow HX"
LMTD_cflow = (DELTAT_1 - DELTAT_2)/ln(DELTAT_1/DELTAT_2) "C...determines LMTD"
Q = m_h * cp_h * (T_h_i - T_h_o) "W...heat tr."
Q = U * A_cflow * LMTD_cflow "Finds A_cflow for counter-flow HX"
"LMTD for a parallel flow HX:"
DT_1 = T_h_i - T_c_i " Temp diff at inlet of HX --- for parallel flow HX"
DT_2 = T_h_o - T_c_o " Temp diff at exit of HX --- for parallel flow HX"
LMTD_pflow = (DT_1 - DT_2)/ln(DT_1/DT_2) "C...determines LMTD"
Q = U * A_pflow * LMTD_pflow "Finds A_pflow for parallel-flow HX"
Results:
Thus:
Exit temp of water = 37.16 C … Ans.
Area for parallel flow HX = 135.8 m^2 …. Ans.
Area for counter-flow HX = 80.92 m^2 …. Ans.
--------------------------------------------------------------------------------------------------------------
16. Prob. B.2.2. Sat. steam at 120 C is condensing on the outer surface of a single pass HX. The overall heat
transfer coeff is 1600 W/m^2.C. Determine the surface area of the HX required to heat 2000 kg/h of water
from 20 C to 90 C. Also determine the rate of condensation of steam (kg/h). Assume latent heat of steam
as 2195 kJ/kg. [VTU-Aug. 2001]
Fig. Prob.B.2.2.
EES Solution:
"Data:"
m_c = 2000 [kg/h] * convert (kg/h, kg/s) "....cold fluid -- water"
T_h = 120 [C]
T_c_i = 20 [C]
T_c_o = 90 [C]
cp_c = 4180 [J/kg-C]
U= 1600 [W/m^2-C]
"LMTD for a condenser:"
DELTAT_1 = T_h - T_c_i " Temp diff at inlet "
DELTAT_2 = T_h - T_c_o " Temp diff at exit "
LMTD = (DELTAT_1 - DELTAT_2)/ln(DELTAT_1/DELTAT_2) "C...determines LMTD"
Q = m_c * cp_c * (T_c_o - T_c_i) "W....finds heat tr, Q"
Q = U * A * LMTD "Finds Area, A for condenser"
h_fg = 2195000 [J/kg] "...latent heat for steam condensing"
m_h * h_fg = Q " kg/s...determines mass of steam condensed"
m_steam_perhour = m_h * convert (kg/s, kg/h)
Results:
17. Thus:
Area of HX = A = 1.747 m^2 …. Ans.
Rate of condensation of steam = 266.6 kg/h … Ans.
"Prob. B.2.3. A simple HX consisting of two concentric passages is used for heating 1110 kg/h of oil (cp
= 2.1 kJ/kg.K) from 27 C to 49 C. The oil flows through the inner pipe made of copper (OD = 2.86 cm,
ID = 2.54 cm, k = 350 W/m.K) and the surface heat transfer coeff on the oil side is 635 W/m^2.K. The oil
is heated by hot water supplied at a rate of 390 kg/h with an inlet temp of 93 C. The water side heat
transfer coeff. is 1270 W/m^2.K. The fouling factors on the oil and water sides are 0.0001 and 0.0004
m^2.K/W respectively. What is the length of HX required for (i) parallel flow, and
(ii) counter-flow ? [VTU-Jan-Feb.- 2006]"
Fig. Prob.B.2.3(a)..Counter-flow arrangement
Fig. Prob.B.2.3(b)..Parallel flow arrangement
18. EES Solution:
"Data:"
m_h = 390 [kg/h] * convert (kg/h, kg/s) "...hot fluid---water"
m_c = 1110 [kg/h] * convert (kg/h, kg/s) "....cold fluid -- oil - flows through inner pipe"
T_h_i = 93 [C]"...inlet temp of hoy fluid"
T_c_i = 27 [C]"...inlet temp of cold fluid"
T_c_0 = 49[C] "...outlet temp of cold fluid"
cp_h = 4180 [J/kg-C]
cp_c = 2100 [J/kg-C]
h_h = 1270[W/m^2-C]
h_c = 635[W/m^2-C]
d_i = 0.0254[m]
d_0 = 0.0286[m]
k_wall = 350 [W/m-C]
R_fi = 0.0001[m^2-C/W]
R_f0 = 0.0004[m^2-C/W]
L = 1[m]"...assumed"
A_i = pi * d_i * L "[m^2]....inside surface area"
A_0 = pi * d_0 * L "[m^2]....outside surface area"
"Calculations:"
"Overall heat tr coeff."
CALL Overall_U_cyl ( L, k_wall, d_i, d_0, h_c, h_h, R_fi, R_f0: R_cond, R_i, R_0, R_ci, R_c0, R_tot, U_i,
U_0) "..here, we get U_i and U_0 "
Running the above program, we get U_i:
i.e. U_i = 365.8 W/m^2.C .. Ans.
Continuing the program:
m_h * cp_h * (T_h_i - T_h_o) = m_c * cp_c * (T_c_o - T_c_i) " C...determines T_h_o"
19. "LMTD for a counter-flow HX:"
DELTAT_1 = T_h_i - T_c_o " Temp diff at inlet of HX --- for counter flow HX"
DELTAT_2 = T_h_o - T_c_i " Temp diff at exit of HX --- for counter flow HX"
LMTD_cflow = (DELTAT_1 - DELTAT_2)/ln(DELTAT_1/DELTAT_2) "C...determines LMTD"
Q = m_h * cp_h * (T_h_i - T_h_o) "W...total heat tr."
Q = U_i * A_cflow * LMTD_cflow "Finds A_cflow for counter-flow HX"
A_cflow = pi* d_i * L_cflow "finds L_cflow, Length for cflow HX"
"LMTD for a parallel flow HX:"
DT_1 = T_h_i - T_c_i " Temp diff at inlet of HX --- for parallel flow HX"
DT_2 = T_h_o - T_c_o " Temp diff at exit of HX --- for parallel flow HX"
LMTD_pflow = (DT_1 - DT_2)/ln(DT_1/DT_2) "C...determines LMTD"
Q = U_i * A_pflow * LMTD_pflow "Finds A_pflow for parallel-flow HX"
A_pflow = pi* d_i * L_pflow "finds L_pflow, Length for pflow HX"
Results:
20. Thus:
Length of HX for parallel flow HX = L_pflow = 15.16 m …. Ans.
Length of HX for counter-flow HX = L_cflow = 12.49 m … Ans.
---------------------------------------------------------------------------------------------------------
!*G 5 H* # ; +
"
Let us write EES PROCEDURE to find LMTD_CF and F for a Shell & Tube HX:
We recollect:
Equations:
If R is not equal to 1:
If R = 1:
21. In the above, N is the no. of simple shells or no. of shell passes.
-----------------------------------------------------------------------------------------------------
Following is the EES Procedure:
$UnitSystem SI Pa C J
PROCEDURE Shell_and_TubeHX_LMTD_F(Tshell_1,Tshell_2,Ttube_1,Ttube_2,N :
R,P,F,LMTD_CF,LMTD_corrected)
"Gives R, P, F, LMTD_CF and LMTD_corrected as output:"
"Input: Inlet and exit temps of Shell side and Tube side fluids, and N is the no. of simple shells or no. of
Shell passes"
DT1 := Tshell_1 - Ttube_2
DT2 := Tshell_2 - Ttube_1
LMTD_CF := ABS(DT1 - DT2) / ABS (ln (DT1/DT2))
P := (Ttube_2 - Ttube_1) / (Tshell_1 - Ttube_1)
R := (Tshell_1 - Tshell_2) / (Ttube_2 - Ttube_1)
IF (Tshell_1 = Tshell_2) OR (Ttube_1 = Ttube_2) THEN
F := 1
LMTD_corrected := F * LMTD_CF
RETURN
ENDIF
IF ( R = 1) THEN
X := P / (N - N * P + P)
F := (X * sqrt(2)) / ((1 - X) * ln( (2 * (1 - X) + x * sqrt(2)) / (2 * (1 - X) - x * sqrt(2))))
ENDIF
IF ( R <> 1) THEN
X :=(1 - ((R * P -1) / ( P - 1))^ (1/N)) / ( R - ((R * P -1)/(P - 1))^(1/N))
F := (sqrt(R^2 + 1) / ( R - 1)) * ln ((1 - X) / (1 - R * X)) / ln ((2/X - 1 - R + sqrt(R^2 + 1)) / (2/X - 1 - R -
sqrt(R^2 + 1)))
ENDIF
LMTD_corrected := F * LMTD_CF
22. END
"---------------------------------------------------------------------"
Let us use this PROCEDURE to solve the following problem:
Prob. B.2.4. In a Shell & Tube HX, water, making one Shell pass, at a rate of 1 kg/s is heated from 35 to
75 C by an oil of sp. heat 1900 J/kg.C. Oil flows at a rate of 2.5 kg/s through the tubes making 2 passes
and enters the Shell at 110 C. If the overall heat transfer coeff. U is 350 W/m^2.C, calculate the area
required."
Fig. Prob.B.2.4. Temp profile for Counter-flow arrangement
EES Solution:
"Data:"
Tshell_1 = 35[C]"....inlet temp of water"
Tshell_2 = 75 [C]"....exit temp of water"
Ttube_1 =110 [C]"....inlet temp of oil"
m_water = 1 [kg/s]
cp_water = 4180 [J/kg-C]
m_oil = 2.5 [kg/s]
cp_oil = 1900 [J/kg-C]
U = 350 [W/m^2-C]
N = 1
"Calculation:"
"Total heat transferred, Q:"
Q = m_water * cp_water * (Tshell_2 - Tshell_1) "[W]"
Q = m_oil * cp_oil * (Ttube_1 - Ttube_2)"[C]...finds exit temp of oil, Ttube_2"
"Also:"
Q = U * A * F * LMTD_CF "...where F is the LMTD correction factor, and LMTD_CF is the LMTD for a true
counter-flow HX"
23. "Get F and LMTD by calling the EES PROCEDURE written above:"
CALL Shell_and_TubeHX_LMTD_F(Tshell_1,Tshell_2,Ttube_1,Ttube_2,N :
R,P,F,LMTD_CF,LMTD_corrected)
Results:
Thus:
F = 0.8 … LMTD correction factor
A = 15.99 m^2 …. Area required for the HX …. Ans.
Note: Since values of R and P are also returned by the program, check the value of F from the graph
(given at the beginning of this chapter, reproduced below).
26. !*G 5 H* # ; L
# M #N, # M #N"
$UnitSystem SI Pa C J
PROCEDURE Crosssflow_OneMixed_LMTD_F(Tmix_1,Tmix_2,Tunmix_1,Tunmix_2,C_mix,C_unmix :
LMTD_CF,P,R,F,LMTD_corrected)
"Gives P, R, F, LMTD_CF and LMTD_corrected as output:
F is the LMTD correction factor. R and P are also output; so, F can be verified from the chart too."
"Input: Inlet and exit temps of mixed and unmixed fluids, andCmix and Cunmix are capacity rates of
respective fluids"
DT1 := Tmix_1 - Tunmix_2
DT2 := Tmix_2 - Tunmix_1
LMTD_CF := ABS(DT1 - DT2) / ABS (ln (DT1/DT2))
P := (Tunmix_2 - Tunmix_1) / (Tmix_1 - Tunmix_1)
R := (Tmix_1 - Tmix_2) / (Tunmix_2 - Tunmix_1)
IF (Tmix_1 = Tmix_2) OR (Tunmix_1 = Tunmix_2) THEN
F := 1
LMTD_corrected := F * LMTD_CF
RETURN
ENDIF
IF ( C_unmix < C_mix) THEN
C := C_unmix/C_mix
epsilon := ABS((Tunmix_2 - Tunmix_1) / (Tmix_1 - Tunmix_1))
27. NTU_counterflow := (1 / (C-1)) * ln((epsilon - 1) / (C * epsilon - 1))
NTU_crossflow := -ln(1 + (1/C) * ln (1 - epsilon * C))
F := NTU_counterflow / NTU_crossflow
LMTD_corrected := F * LMTD_CF
ENDIF
IF ( C_mix <= C_unmix) THEN
C := C_mix/C_unmix
epsilon := ABS((Tmix_2 - Tmix_1) / (Tmix_1 - Tunmix_1))
IF (C = 1) THEN NTU_counterflow := epsilon / (1 - epsilon)
IF (C < 1) THEN NTU_counterflow := (1 / (C-1)) * ln((epsilon - 1) / (C * epsilon - 1))
NTU_crossflow := (-1 / C) * ln(C * ln(1 - epsilon) + 1)
F := NTU_counterflow / NTU_crossflow
LMTD_corrected := F * LMTD_CF
ENDIF
END
"---------------------------------------------------------------------"
Let us work out a problem to demonstrate the use of this Function for F for a cross flow
HX with one fluid un-mixed:
"Prob. B.2.5. Consider a cross flow HX in which oil (cp = 1900 J/kg.K) flowing inside tubes is heated
from 15 C to 85 C by steam blowing across the tubes. Steam enters at 130 C and leaves at 110 C, with a
mass flow rate of 5.2 kg/s. Overall heat transfer coeff, U is 275 W/m^2.K. For steam, cp = 1860 J/kg.K.
Calculate the surface area of the HX.”
Fig. Prob.B.2.5
28. EES Solution:
"Data:"
Tmix_1 = 130 [C]
Tmix_2 = 110 [C]
Tunmix_1 = 15 [C]
Tunmix_2 = 85 [C]
cp_oil = 1900 [J/kg-C]
cp_steam = 1860 [J/kg-C]
m_steam = 5.2 [kg/s]
U = 275 [W/m^2 - C]
"Calculations:"
C_mix = m_steam * cp_steam "W/C... capacity rate of mixed fluid, i.e. steam"
Q = C_mix * (Tmix_1 - Tmix_2) "W"
C_unmix = Q / (Tunmix_2 - Tunmix_1) "W/C ...capacity rate of un-mixed fluid, i.e. oil"
"Using the EES Procedure written earlier:"
CALL Crosssflow_OneMixed_LMTD_F(Tmix_1,Tmix_2,Tunmix_1,Tunmix_2,C_mix,C_unmix :
LMTD_CF,P,R,F,LMTD_corrected)
Results:
Thus:
LMTD correction factor, F = 0.9469
Area of HX, A = 11.1 m^2 …. Ans.
Note: Verify value of F from the graph given above.
=====================================================================
29. !*G 5 H* # ; L
# M #N"
We follow the following procedure:
We digitize each of the above graphs, i.e. for R = 4, 3, 2…0.2. And then, we curve fit each graph with
CurveExpert software and find the equation for each graph. Then, we write the EES Procedure to get F
for given P and R:
As an example, we show the curve fit eqn for R = 4:
30.
31. Now, write the EES Function:
$UnitSystem SI Pa C J
PROCEDURE Crosssflow_BothUnmixed_LMTD_F(T_1,T_2,tt_1,tt_2,C_T,C_tt :
LMTD_CF,P,R,F,LMTD_corrected)
"Gives P, R, F, LMTD_CF and LMTD_corrected as output:
F is the LMTD correction factor. R and P are also output; so, F can be verified from the chart too."
"Input: Inlet and exit temps of fluid 1 (T) and fluid 2 (tt), and C_T and C_tt are capacity rates of respective
fluids"
P := (tt_2 - tt_1) / (T_1 - tt_1)
R := (T_1 - T_2) / (tt_2 - tt_1)
IF (R > 4) OR (R < 0.2) THEN
CALL ERROR ('R should be between 4 and 0.2. Here R is XXXF4', R)
ENDIF
DT1 := T_1 - tt_2
DT2 := T_2 - tt_1
LMTD_CF := ABS(DT1 - DT2) / ABS (ln (DT1/DT2))
IF (T_1 = T_2) OR (tt_1 = tt_2) THEN
F := 1
LMTD_corrected := F * LMTD_CF
RETURN
ENDIF
"For R = 4:"
a40 := 0.9672809; b40 := -4.0433204; c40 := -4.0064623; d40 := -0.3019396
F40 := (a40 + b40 * P)/(1+c40 * P + d40 * P^2)
"For R = 3:"
a30 :=1.0027344; b30 :=-2.9499251; c30 :=-2.8176633; d30 :=-0.059473011
F30 := (a30 + b30 * P)/(1+c30 * P + d30 * P^2)
"For R = 2:"
a20 :=1.0528796; b20:=-2.0982225; c20 :=-1.588803; d20 :=-0.66832947
32. F20 := (a20 + b20 * P)/(1+c20 * P + d20 * P^2)
"For R = 1.5:"
a15 :=1.013584; b15 :=-1.4982864; c15 :=-1.3928085; d15 :=0.1022294
F15 := (a15 + b15 * P)/(1+c15 * P + d15 * P^2)
"For R = 1:"
a10 :=1.0292657; b10 :=-0.23963952; c10 :=0.16670566; d10 :=0.31157928; e10 :=-1.4227221
F10 := (a10 + b10 * P + c10 * P^2 + d10 * P^3 + e10 * P^4)
"For R = 0.8:"
a08 :=1.007952; b08 :=-0.99036246; c08 :=-0.96731788; d08 :=0.14733358
F08 := (a08 + b08 * P)/(1+c08 * P + d08 * P^2)
"For R = 0.6:"
a06 :=1.0510175; b06 :=-1.0614768; c06 :=-0.84847439; d06 :=-0.099409186
F06 := (a06 + b06 * P)/(1+c06 * P + d06 * P^2)
"For R = 0.4:"
a04 :=1.0338023; b04 :=-1.0158388; c04 :=-0.87973396; d04 :=-0.061943671
F04 := (a04 + b04 * P)/(1+c04 * P + d04 * P^2)
"For R = 0.2:"
a02 :=1.016908; b02 :=-0.99564685; c02 :=-0.91765904; d02 :=-0.043092966
F02 := (a02 + b02 * P)/(1+c02 * P + d02 * P^2)
IF ((R >= 0.2) AND (R < 0.4)) THEN
F := F02 + (R - 0.2) * (F04 - F02) / (0.4 - 0.2)
ENDIF
IF ((R >= 0.4) AND (R < 0.6)) THEN
F := F04 + (R - 0.4) * (F06 - F04) / (0.6 - 0.4)
ENDIF
IF ((R >= 0.6) AND (R < 0.8)) THEN
F := F06 + (R - 0.6) * (F08 - F06) / (0.8 - 0.6)
ENDIF
33. IF ((R >= 0.8) AND (R < 1)) THEN
F := F08 + (R - 0.8) * (F10 - F08) / (1 - 0.8)
ENDIF
IF ((R >= 1) AND (R < 1.5)) THEN
F := F10 + (R - 1) * (F15 - F10) / (1.5 - 1)
ENDIF
IF ((R >= 1.5) AND (R < 2)) THEN
F := F15 + (R - 1.5) * (F20 - F15) / (2 - 1.5)
ENDIF
IF ((R >= 2) AND (R < 3)) THEN
F := F20 + (R - 2) * (F30 - F20) / (3 - 2)
ENDIF
IF ((R >= 3) AND (R <= 4)) THEN
F := F30 + (R - 3) * (F40 - F30) / (4 - 3)
ENDIF
LMTD_corrected := F * LMTD_CF
END
"============================================================="
"Prob. B.2.6. A cross-flow HX in which both fluids are un-mixed is used to heat water with engine oil.
Water enter at 30 C and leaves at 85 C at a rate of 1.5 kg/s, while the engine oil with cp = 2.3 kJ/kg.C
enters at 120 C with a mass flow rate of 3.5 kg/s. The heat transfer surface area is 30 m^2. Calculate the
overall heat transfer coefficient by using the LMTD method. [VTU -June - July 2009]"
Fig. Prob.B.2.6.
Now, we shall solve this problem using the EES Function written above:
34. EES Solution:
"Data:"
m_h = 3.5 [kg/s] "...hot fluid---oil"
m_c = 1.5 [kg/s] "....cold fluid -- water"
T_h_i = 120 [C]
T_c_i = 30 [C]
T_c_o = 85 [C]
cp_h = 2300 [J/kg-C]
cp_c = 4180 [J/kg-C]
A = 30 [m^2]
m_h * cp_h * (T_h_i - T_h_o) = m_c * cp_c * (T_c_o - T_c_i) " C...determines T_h_o"
C_h = m_h * cp_h " W/C ... capacity ratre of hot fluid"
C_c = m_c * cp_c " W/C ... capacity ratre of cold fluid"
"To find LMTD Correction factor F for a cross-flow HX....
Either from the graph for a single pass HX with both fluids unmixed:, OR:
Use the EES Function written above"
CALL Crosssflow_BothUnmixed_LMTD_F(T_h_i,T_h_0,T_c_i,T_c_0,C_h,C_c :
LMTD_CF,P,R,F,LMTD_corrected)
Q = m_h * cp_h * (T_h_i - T_h_0) "W...heat tr."
Q = U * A * LMTD_corrected "... Finds U"
"=============================================="
{Note: F = 0.87 approx. " From graph given above, for values of R and P given below:"}
Results:
Thus:
35. LMTD correction factor, F = 0.8721 … Ans.
Overall heat transfer coeff. U = 323.2 W/m^2.C….Ans.
======================================================================
Consider following extension to the above problem:
If oil flow rate (mh) varies from 2.5 kg/s to 5.25 kg/s, with the temperatures Th_i, Tc_i and
Tc_0 remaining const., plot the variation of Th_0, F and U with mh:
First, construct the Parametric Table:
Now, plot the graphs:
36.
37. 3 1 % MD H D H E
O N #"
# 5 L"
"Function to determine Effectiveness of Counterflow HX:"
FUNCTION Epsilon_CountreflowHX(NTU,C_r)
"Finds the effectiveness of Counterflow HX:
Input: NTU, Capacity ratio, C _r( = C_min/C_max)
Output: Effectiveness"
IF (C_r = 0) THEN
Epsilon_CountreflowHX =1 - exp(-NTU)
RETURN
ENDIF
IF (C_r = 1) THEN
Epsilon_CountreflowHX = NTU / (1 + NTU)
RETURN
ENDIF
IF ((C_r > 0) OR (C_r <1)) THEN
Epsilon_CountreflowHX = (1 - exp ( - NTU * (1 - C_r)))/(1 - C_r * exp( - NTU * (1 - C_r))) "Effectiveness
of Counterflow HX."
ENDIF
END
"======================================================"
# ! L"
"Function to determine Effectiveness of Parallel flow HX:"
FUNCTION Epsilon_ParallelflowHX(NTU,C_r)
"Finds the effectiveness of Parallelflow HX:
Input: NTU, Capacity ratio, C _r( = C_min/C_max)
38. Output: Effectiveness"
IF (C_r = 0) THEN
Epsilon_ParallelflowHX =1 - exp(-NTU)
RETURN
ENDIF
IF ((C_r > 0) OR (C_r <=1)) THEN
Epsilon_ParallelflowHX = (1 - exp ( - NTU * (1 + C_r)))/(1 + C_r) "Effectiveness of Parallelflow HX."
ENDIF
END
"======================================================"
"Prob. B.3.1. A water to water HX of a counter-flow arrangement has a heating surface area of 2 m^2.
Mass flow rates of hot and cold fluids are 2000 kg/h and 1500 kg/h respectively. Temperatures of hot and
cold fluids at inlet are 85 C and 25 C respectively. Determine the amount of heat transferred from hot to
cold water and their temps at the exit if the overall heat transfer coeff. U = 1400 W/m^2.K. [VTU -May -
June 2010]:
(b) Plot the variation of Q, Th_o and Tc_o as the mass flow rate of cold fluid, m_c varies from 0.1 to 0.5
kg/s. Assume other conditions to remain the same.”
Fig. Prob.B.3.1. Counter-flow arrangement
EES Solution:
"Data:"
m_h = 2000 [kg/h]* convert(kg/h, kg/s)
m_c = 1500 [kg/h] * convert(kg/h, kg/s)
T_h_i = 85 [C]
39. T_c_i = 25 [C]
cp_h = 4180 [J/kg-C]
cp_c = 4180 [J/kg-C]
U = 1400 [W/m^2-C]
A = 2 [m^2]
"Calculations:"
C_h = m_h * cp_h "W/C...=2322"
C_c = m_c * cp_c "W/C... = 1742 "
C_min = C_c
C_max = C_h
C_r = C_min/C_max
NTU = U * A/C_min "...calculates NTU"
"For counterflow HX: Use the EES Function for Effectiveness of Counterflow HX, written above"
epsilon = Epsilon_CountreflowHX(NTU,C_r) "Effectiveness of Counterflow HX."
"Also:"
epsilon = (T_c_0 - T_c_i) / (T_h_i - T_c_i) "...by definition, taking the min fluid... calculates Tc_0"
Q = m_h * cp_h * (T_h_i - T_h_0) "W...heat tr... calculates Th_0"
Q= m_c * cp_c * (T_c_0 - T_c_i) "W... calculates Q"
"============================================================="
Results:
Thus:
Q = 69418 W, Th_0 = 55.11 C, Tc_0 = 64.86 C .. Ans.
-----------------------------------------------------------------------------------------------------
Plot the variation of Q, Th_o and Tc_o as the mass flow rate of cold fluid, m_c varies from
0.1 to 0.5 kg/s. Assume other conditions to remain the same:
First, compute the Parametric Table:
41. “Prob. B.3.2. (M.U. 1995): Consider a heat exchanger for cooling oil which enters at 180 C, and cooling
water enters at 25 C. Mass flow rates of oil and water are: 2.5 and 1.2 kg/s, respectively. Area for heat
transfer = 16 m2. Sp. heat data for oil and water and overall U are given: Cpoil = 1900 J/kg.K; Cpwater =
4184 J/kg.K; U = 285 W/m2.K. Calculate outlet temperatures of oil and water for Parallel and counter-
flow HX.”
"
"Data:"
m_h = 2.5 [kg/s]
m_c = 1.2 [kg/s]
T_h_i = 180 [C]
T_c_i = 25 [C]
cp_h = 1900 [J/kg-C]
cp_c = 4184 [J/kg-C]
U = 285 [W/m^2-C]
A = 16 [m^2]
"Calculations:"
C_h = m_h * cp_h "W/C...=4750"
C_c = m_c * cp_c "W/C... = 5021 "
C_min = C_h
C_max = C_c
C_r = C_min/C_max
NTU = U * A/C_min "...calculates NTU"
"For Parallel flow HX: Using the EES Function for effectiveness:"
epsilon_pflow = Epsilon_ParallelflowHX(NTU,C_r) "...finds effectiveness of parallel flow HX"
"But, since hot fluid is 'minimum fluid', we have:"
epsilon_pflow = (T_h_i - T_h_0) / (T_h_i - T_c_i) "... finds T_h_0, exit temp of hot fluid"
Q = C_h * (T_h_i - T_h_0) "W ... heat transferred"
Q = C_c * (T_c_0 - T_c_i) "... finds T_c_0, exit temp of cold fluid"
Results for Parallel flow HX:
42. Thus:
For Parallel flow HX: T_h_0 = 112.6 C, T_c_0 = 88.72 C, Q = 319915 W …. Ans.
For Counter flow HX:
Add the following part, after commenting out the portion for parallel flow HX, in the above EES program:
"For Counter flow HX: Using the EES Function for effectiveness:"
epsilon_cflow = Epsilon_CountreflowHX(NTU,C_r) "Effectiveness of Counterflow HX."
"But, since hot fluid is 'minimum fluid', we have:"
epsilon_cflow = (T_h_i - T_h_0) / (T_h_i - T_c_i) "... finds T_h_0, exit temp of hot fluid"
Q = C_h * (T_h_i - T_h_0) "W ... heat transferred"
Q = C_c * (T_c_0 - T_c_i) "... finds T_c_0, exit temp of cold fluid"
"------------------------------------------------------------------------------------------------"
Results for Counter flow HX:
Thus:
For Counter flow HX: T_h_0 = 103.1 C, T_c_0 = 97.78 C, Q = 365397 W …. Ans.
# + L, ( $ #
&,=,)J $ "
FUNCTION Epsilon_ShellAndTube_HX(NTU,C_r)
"Finds the effectiveness of Shell & Tube HX with - One shell Pass, 2,4,6..... Tube passes:
Input: NTU, Capacity ratio, C _r( = C_min/C_max)
Output: Effectiveness"
IF (C_r = 0) THEN
43. Epsilon_ShellAndTube_HX :=1 - exp(-NTU)
RETURN
ENDIF
IF ((C_r > 0) OR (C_r <=1)) THEN
AA := 1 + exp(-NTU * (1 + C_r^2)^0.5)
BB := 1 - exp(-NTU * (1 + C_r^2)^0.5)
Epsilon_ShellAndTube_HX := 2 * (1 + C_r + (1 + C_r^2)^0.5 * AA / BB)^(-1) "Effectiveness of Shell &
Tube HX with one shell pass, 2,4,6... tube passes."
ENDIF
END
"======================================================"
"Prob. B.3.3: Hot oil at a temperature of 180 C enters a Shell and tube HX and is cooled by water
entering at 25 C. There is one shell pass and 6 tube passes in the HX and the overall heat transfer coeff. is
350 W/(m2.K). Tube is thin-walled, 15 mm ID and length per pass is 5 m. Water flow rate is 0.3 kg/s and
oil flow rate is 0.4 kg/s. Determine the outlet temperatures of oil and water and also the heat transfer rate
in the HX. Given: sp. heat of oil = 1900 J/(kg.K) and sp. heat of water = 4184 J/(kg.K)"
3 1 1 + L $ # $
Solution:
"Data:"
44. m_h = 0.4 [kg/s]
m_c = 0.3 [kg/s]
Th_1 = 180 [C]
Tc_1 = 25 [C]
U = 350 [W/m^2 - C]
D = 0.015 [m] "..ID of tubes"
L = 5 [m] "...length per pass"
cp_h = 1900 [J/kg-C]
cp_c = 4184 [J/kg-C]
n = 6 "...no. of tube passes"
"Calculations:"
C_h = m_h * cp_h " W/C ... capacity rate of hot fluid = 760 W/C"
C_c = m_c * cp_c " W/C ... capacity rate of cold fluid = 1255 W/C "
"Therefore: hot fluid is the 'minimum fluid"
C_min = C_h
C_max = C_c
C_r = C_min / C_max "... capacity ratio"
"Total heat transfer area, A:"
A = n * pi * D * L "m^2... n = 6. no. of tube passes"
NTU = U * A / C_min "... calculates NTU"
"Knowing NTU and C_r, calculate effectiveness ... Use the EES Function written above:"
epsilon = Epsilon_ShellAndTube_HX(NTU,C_r) ".... calculates effectiveness"
Q_max = C_min * (Th_1 - Tc_1) " W ... max possible heat transfer"
Q_actual = epsilon * Q_max " W... actual heat transfer in the HX"
"Outlet temps of fluids:"
Q_actual = C_h * (Th_1 - Th_2) "... calculates outlet temp of hot fluid, Th_2 (deg.C)"
Q_actual = C_c * (Tc_2 - Tc_1) "... calculates outlet temp of cold fluid, Tc_2 (deg.C)"
45. * "
Thus:
Th_2 = 115.7 C, Tc_2 = 63.91 C, Q_actual = 48836 W … Ans.
Plot Q_actual, Th_2 and Tc_2 as m_h varies from 0.2 to 0.65 kg/s, other quantities
remaining unchanged:
First, compute the Parametric Table:
Now, plot the results:
46.
47. # 5 L, #
M #N"
"Function to determine Effectiveness of Crossflow HX with both fluids Unmixed:"
FUNCTION Epsilon_Crossflow_bothUnmixed(NTU,C_r)
"Finds the effectiveness of Crossflow HX with both fluids Unmixed:
Input: NTU, Capacity ratio, C _r( = C_min/C_max)
Output: Effectiveness"
IF (C_r = 0) THEN
Epsilon_Crossflow_bothUnmixed :=1 - exp(-NTU)
RETURN
ENDIF
IF ((C_r > 0) OR (C_r <=1)) THEN
Epsilon_Crossflow_bothUnmixed := 1 - exp ( (1/C_r) * NTU^0.22 * ( exp( - C_r * NTU^0.78) - 1))
"Effectiveness of Cross-flow HX ..Ref: Incropera. "
ENDIF
END
"======================================================"
Now, solve a problem using the above EES Function:
"Prob.B.3.4. A cross-flow HX (both fluids unmixed), having a heat transfer area of 8.4 m^2, is to heat air
(cp = 1005 J/kg.K) with water (cp = 4180 J/kg.K). Air enters at 18 C with a mass flow rate of 2 kg/s while
water enters at 90 C with a mass flow rate of 0.25 kg/s. Overall heat transfer coeff is 250 W/m^2.K
Calculate the exit temps of the two fluids and the heat transfer rate. [VTU - July-Aug. 2004]"
(b) Plot the variation of Q and Th_o and Tc_o as air flow rate, m_c varies from 1.5 to 3 kg/s, all other
conditions remaining the same as earlier:
Fig. Prob.B.3.4. Cross-flow arrangement
48. EES Solution:
"Data:"
m_h = 0.25 [kg/s] "....water is the hot fluid"
m_c = 2[kg/s] "....air is the cold fluid"
T_h_i = 90 [C]
T_c_i = 18 [C]
cp_c = 1005 [J/kg-C]
cp_h = 4180 [J/kg-C]
U = 250 [W/m^2-C]
A = 8.4 [m^2]
Calculations:
C_h = m_h * cp_h "W/C...= 1045 … capacity rate of hot fluid"
C_c = m_c * cp_c "W/C... = 2010 … capacity rate of cold fluid"
C_min = C_h “….min. capacity rate”
C_max = C_c ”…max. capacity rate”
C_r = C_min/C_max “… Capacity ratio”
NTU = U * A /C_min “…NTU by definition”
"For cross-flow HX: Using the EES Function for effectiveness of Cross flow HX, with both fluids Unmixed,
written earlier:"
epsilon = Epsilon_Crossflow_bothUnmixed(NTU,C_r) "Effectiveness of Cross-flow HX ..Ref: Incropera. "
"Also:"
epsilon = (T_h_i - T_h_o)/(T_h_i - T_c_i) "...determines exit temp of hot fluid, i.e. water, T_h_o"
Q= m_h * cp_h * (T_h_i - T_h_o) "W ... determines Q"
Q = m_c * cp_c * (T_c_o - T_c_i) "W...determines exit temp of cold fluid, i.e. air, T_c_o."
"====================================================================="
Results:
Thus:
Q = 55286 W … heat transferred … Ans.
Th_0 = 37.09 C … exit temp of hot fluid (water) …. Ans.
49. Tc_0 = 45.51 C … exit temp of cold fluid (air) … Ans.
---------------------------------------------------------------------------------------------------------
(b) Plot the variation of Q and Th_o and Tc_o as air flow rate, m_c varies from 1.5 to 3
kg/s, all other conditions remaining the same as earlier:
First, compute the Parametric Table:
Next, plot the results:
50. =====================================================================
# 5 L, #
MH #N, # # M #N"
FUNCTION Epsilon_Crossflow_OneUnmixed(NTU,C_mixed, C_unmixed)
"Finds the effectiveness of Crossflow HX with one fluid 'Unmixed', and the other, 'mixed':
Input: NTU, Capacity rates, C_mixed and C_unmixed .
Output: Effectiveness"
IF (C_mixed > C_unmixed ) THEN
C_r := C_unmixed / C_mixed
IF (C_r = 0) THEN
Epsilon_Crossflow_OneUnmixed := 1 - exp (-NTU)
RETURN
ENDIF
AA := -C_r * (1 - exp(-NTU))
BB = 1 - exp(AA)
Epsilon_Crossflow_OneUnmixed := (1/ C_r) * BB
ENDIF
IF (C_unmixed >= C_mixed ) THEN
51. C_r := C_mixed / C_unmixed
IF (C_r = 0) THEN
Epsilon_Crossflow_OneUnmixed := 1 - exp (-NTU)
RETURN
ENDIF
AA := (-1/C_r) * ( 1 - exp(-NTU * C_r))
Epsilon_Crossflow_OneUnmixed := 1 - exp( AA) "Effectiveness of Cross-flow HX ..Ref: Incropera. "
ENDIF
END
"======================================================"
Now, solve a problem using the above EES Function:
Prob. B.3.5. Consider a cross flow HX where oil flowing through the tubes is heated by steam flowing
across the tubes. Oil (cp = 1900 J/kg.C) is heated from 15 C to 85 C and steam (cp = 1860 J/kg.C) enters
at 130 C and leaves at 110 C with a mass flow rate of 5.2 kg/s. Overall heat transfer coeff U = 275
W/m^2.C. Calculate the surface area required for this HX.
52. EES Solution:
"Data:"
"Steam is the 'mixed' fluid and oil is the 'un-mixed' fluid."
m_h = 5.2 [kg/s] "....steam is the hot fluid, 'mixed' fluid"
T_h_i = 130 [C]
T_c_i = 15 [C]
T_c_0 = 85 [C]
T_h_0 = 110 [C]
cp_c = 1900 [J/kg-C]
cp_h = 1860 [J/kg-C]
U = 275 [W/m^2-C]
"Calculations:"
C_mixed = m_h * cp_h "W/C...= 9672 capacity rate of hot fluid"
Q = C_mixed * (T_h_i - T_h_0) "W ... heat transfer in HX"
Q = C_unmixed * (T_c_0 - T_c_i) "...finds C_unmixed, ... 2763 [W/C]"
C_unmixed = m_c * cp_c "...finds m_c, flow rate of cold fluid (oil), kg/s"
"Since we find that oil is the minimum fluid, we can write for Effectiveness:"
epsilon = (T_c_0 - T_c_i) / (T_h_i - T_c_i) "... effectivenes, by definition"
"Then, to find NTU: Use the EES Function written above for Effectiveness of this HX.
Note that when effectiveness and Capacity rates are known, the same Function determines the
other unknown, viz. NTU:"
epsilon = Epsilon_Crossflow_OneUnmixed(NTU,C_mixed, C_unmixed) "Effectiveness of Cross-flow HX
.....finds NTU "
"Also, since oil is the 'minimum' fluid:"
NTU = U * A / C_unmixed "...finds A [m^2]"
"====================================================================="
53. Results:
Thus:
Effectiveness of HX = epsilon = 0.6087 … Ans.
NTU = 1.105 …. Ans.
Q = 193440 W …. Ans.
Area of HX = A = 11.1 m^2 … Ans.
# 5 B4 B6 #
$ L"
Kays and London have studied a large number of compact heat exchanger matrices and
presented their experimental results in the form of generalized graphs. Heat transfer data is
plotted as Colburn j-factor, jH = St.Pr2/3
against Re,
where, St = Stanton number = h/(G.Cp), Pr = Prandtl number = µ.Cp/k, and
Re = G.Dh/µ, G = mass velocity (= mass flow rate/Area of cross-section), kg/(s.m2
.)
In the same graphs, friction factor, f, is also plotted against Re.
One typical graph of characteristics for a plate– finned circular tube matrix (data of Trane Co.) given
by Kays and London is shown below:
54. ( # $ 4 #
In the above graph, Reynolds No. is shown on the x-axis.
Colburn j-factor, jH is used to get heat transfer coeff. h.
Friction factor is used to get the frictional pressure drop:
Writing EES Procedure to get j_H and f for given Reynolds No. (Re):
The procedure is as follows:
Digitize the graphs for jH and f, using PlotDigitizer, a freely available, java based software.
(Ref: http://plotdigitizer.sourceforge.net).
Copy the data generated in to EXCEL, draw the curves and get curve-fit equations in EXCEL:
And, then using the curve-fit equations, write the EES Functions for j_H and f:
From EXCEL:
55.
56. Thus, we have the curve fit equations for j_H and f:
Now, write the EES Procedure to get j_H and f:
"EES Procedure to determine Colburn j-factor (j_H) and friction factor (f) for a compact HX -
a plate- finned circular tube matrix (data of Trane Co.) given by Kays and London:"
PROCEDURE Colburn_jH_and_friction_factor(Re:j_H, f)
"Finds the Colburn j-factor (j_H) and friction factor (f) for a compact HX -
a plate- finned circular tube matrix (data of Trane Co.)
Input: Reynolds No. (Re)
Output: j_H and f"
"Remember: j_H ( = St. Pr^(2/3)) where St = (h / G. Cp), Pr = (mu . Cp / k), Re = (G. Dh / mu)"
IF ((Re < 2000) OR (Re > 12000)) THEN
CALL ERROR ('Re should be between 2000 and 12000 !')
ENDIF
Re_modified := Re / 1000
57. j_H := 0.01052 * (Re_modified)^(-0.40002)
f := 0.02940 * (Re_modified)^(-0.21105)
END
"======================================================"
Now, let us solve a problem on compact heat exchangers:
Prob. B.3.6: Air at 2 atm and 400 K flows at a rate of 5 kg/s, across a finned circular tube matrix, for
which heat transfer and friction factor characteristics are shown in Fig.1 above. Dimensions of the heat
exchanger matrix are: 1 m (W) x 0.6 m (Deep) x 0.5 m (H), as shown in Fig.below. Find: (a) the heat
transfer coeff. (b) the friction factor, and (c) ratio of core friction pressure drop to the inlet pressure.
EES Solution:
"Data:"
m = 5 [kg/s]
A_fr = 0.5 [m^2] "...frontal area"
L = 0.6 [m] "...length of flow"
P_air = 2 * 101.3 * 10^3 [Pa]
T_air = 400 - 273 "[C]"
"Properties of Air at 2 atm and 400 K:"
rho = Density(Air,T=T_air,P=P_air) "[kg/m^3]"
mu = Viscosity(Air,T=T_air)" Dyn. viscosity, kg/m.s"
cp = SpecHeat(Air,T=T_air) "sp. heat, J/kg.C"
Pr = Prandtl(Air,T=T_air) "Prandtl No."
58. "From the Fig.1, i.e. Heat transfer and friction factor characteristics for plate-finned circular tube matrix
heat exchanger, we have:"
sigma = 0.534 ; D_h = 3.63E-03 "m .... hydraulic dia"
sigma = A_min / A_fr " .. by definition"
"Mass velocity: G:"
G = m / A_min " .... mass velocity, [kg/s-m^2]"
"Reynolds No.:"
Re = G * D_h / mu "...Reynolds No."
"We get Re = 2965. Now, use the EES Procedure written above to get j_H and f:"
CALL Colburn_jH_and_friction_factor(Re:j_H, f)
"But, we have:"
j_H = St * Pr^(2/3) "... Colburn j-factor"
St = h /(G * cp)"... Stanton No... h = heat tr coeff. .... finds h"
"Pressure drop:"
" DELTAP_f = f * (G^2 / (2 * rho)) * (A / A_min)
And, (A / A_min) = 4 * L / D_h . Therefore:"
DELTAP_f = f * (G^2 / (2 * rho)) * (4 * L / D_h ) " Pressure drop, N/m^2"
PressureDrop_fraction =( DELTAP_f / P_air) *100 " % . of inlet pressure"
"================================================================"
Results:
59. Thus:
(a) heat transfer coeff. = h = 162.8 W/m^2.C … Ans.
(b) friction factor = f = 0.02337 …. Ans.
(c) ratio of core friction pressure drop to the inlet pressure = 0.7582 % …. Ans.
Plot the variation of h and Pf as mass flow rate varies from 4 to 10 kg/s, other quantities
remaining unchanged:
First, compute the Parametric Table:
Now, plot the results: