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
The aim of this experiment is to measurement linear thermal along z direction conductivity and to investigate and verify Fourier’s Law for linear heat conduction along z direction and we proved that K is inversely proportional with ΔT, and we have many errors in our experiment that made the result not clear.
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
The aim of this experiment is to measurement linear thermal along z direction conductivity and to investigate and verify Fourier’s Law for linear heat conduction along z direction and we proved that K is inversely proportional with ΔT, and we have many errors in our experiment that made the result not clear.
heat conduction and its mechanisms ,thermal conductivity,Fourier law,variation of thermal conductivity with temperature in metals and solids,steady and unsteady states,biot and Fourier numbers and their significance, Lumped heat analysis
Heat transfer enhancement by nanofluid Suhail Patel
The purpose of this paper is to look into the present aspects of “Nanotechnology”. This gives a brief description of how heat transfer enhances using Nanofluid And its application in various fields viz. heat transportation, military applications, medical, etc. This paper focuses one explaining the basic mechanisms of improvement in heat transfer by addition nanoparticles.
Shell and Tube Heat Exchanger in heat TransferUsman Shah
This slide will explain you the chemical engineering terms .Al about the basics of this slide are explain in it. The basics of fluid mechanics, heat transfer, chemical engineering thermodynamics, fluid motions, newtonian fluids, are explain in this process.
This presentation is on shell and tube heat exchanger in which its design parameters and its troubleshooting conditions designed for better understanding and learning of all
heat conduction and its mechanisms ,thermal conductivity,Fourier law,variation of thermal conductivity with temperature in metals and solids,steady and unsteady states,biot and Fourier numbers and their significance, Lumped heat analysis
Heat transfer enhancement by nanofluid Suhail Patel
The purpose of this paper is to look into the present aspects of “Nanotechnology”. This gives a brief description of how heat transfer enhances using Nanofluid And its application in various fields viz. heat transportation, military applications, medical, etc. This paper focuses one explaining the basic mechanisms of improvement in heat transfer by addition nanoparticles.
Shell and Tube Heat Exchanger in heat TransferUsman Shah
This slide will explain you the chemical engineering terms .Al about the basics of this slide are explain in it. The basics of fluid mechanics, heat transfer, chemical engineering thermodynamics, fluid motions, newtonian fluids, are explain in this process.
This presentation is on shell and tube heat exchanger in which its design parameters and its troubleshooting conditions designed for better understanding and learning of all
Modes of Heat Transfer - An Experimental ApproachNoaman Ahmed
Comparison of theoretical and steady-state temperature profiles in metallic rods of Copper, Aluminum, and Iron including the determination of net heat transfer in each metallic rod and the comparison of rate of heat transfer due to conduction, convection, and radiation.
Introduction
Mechanism of Heat Flow
Conduction
Heat Flow through a Cylinder-Conduction
Conduction through fluids
Convection
Film type condensation
Cold liquid-boiling of liquids
Modes of Feed-Heat Transfer
Thermal Radiation
Black Body
Grey body
Equipments
References
2.1 Heat
Heat is a form of energy. According to the principle of thermodynamics whenever a physical or chemical transformation occurs heat flow into or leaves the system.
A number of sources of heat are used for industrial scale operations steam and electric power is the chief sources to transfer heat. It is essential to cover steam without any loses to the apparatus in which it is used. The study of heat transfer processes helps in be signing the plant efficiently and economically
2.2 Heat Transfer:-
Work is one of the basic modes of energy transfer in machines the action of force on a moving body is identified as work. The work is done by a force as it acts upon a body moving in the direction of the force.
Work transfer is considered as occurring between the system and the surroundings work is said to be done by a system is the sole effect on things external to the system can be reduced to the raising of a weight.
If a system has a non-adiabatic boundary its temperature is not independent of the temperature of the surroundings and for the system between the states 1 and 2 the work w depends on path and the differential d-w is inexact. The work depends on the terminal state 1 and 2 as well as non-adiabatic path connecting them. For consistency with the principle of conservation of energy. Some type of energy transfer must have occurred because of the temperature difference between the system and its surroundings and it is identified as heat thus when an effect in a system occurs solely as result of temperature difference between the system and some other system the process in which the effect occur shall be called a transfer of heat from the system at the higher temperature to the system at the lower temperature.
1.1 Evaporation
1.2 Distillation
1.3 Drying
1.4 Crystallization
1.5 Sterilization
Application of Heat Transfer in Pharmaceuticals Industries
Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
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.
Overview of the fundamental roles in Hydropower generation and the components involved in wider Electrical Engineering.
This paper presents the design and construction of hydroelectric dams from the hydrologist’s survey of the valley before construction, all aspects and involved disciplines, fluid dynamics, structural engineering, generation and mains frequency regulation to the very transmission of power through the network in the United Kingdom.
Author: Robbie Edward Sayers
Collaborators and co editors: Charlie Sims and Connor Healey.
(C) 2024 Robbie E. Sayers
Student information management system project report ii.pdfKamal Acharya
Our project explains about the student management. This project mainly explains the various actions related to student details. This project shows some ease in adding, editing and deleting the student details. It also provides a less time consuming process for viewing, adding, editing and deleting the marks of the students.
Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
In this month's edition, along with this month's industry news to celebrate the 13 years since the group was created we have articles including
A case study of the used of Advanced Process Control at the Wastewater Treatment works at Lleida in Spain
A look back on an article on smart wastewater networks in order to see how the industry has measured up in the interim around the adoption of Digital Transformation in the Water Industry.
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.
Learn about the cost savings, reduced environmental impact, and minimal disruption associated with trenchless technology. Discover detailed explanations of popular techniques such as pipe bursting, cured-in-place pipe (CIPP) lining, and directional drilling. Understand how these methods can be applied to various types of infrastructure, from residential plumbing to large-scale municipal systems.
Ideal for homeowners, contractors, engineers, and anyone interested in modern plumbing solutions, this guide provides valuable insights into why trenchless pipe repair is becoming the preferred choice for pipe rehabilitation. Stay informed about the latest advancements and best practices in the field.
Water scarcity is the lack of fresh water resources to meet the standard water demand. There are two type of water scarcity. One is physical. The other is economic water scarcity.
2. ABSTRACT
• Heat transfer is one of the important phenomena in any mechanical device. This report describes the heat
transfer fundamentals and its characteristics. This project is about the experimental investigation of
thermal conductivity of metal rod. In the experimental investigation different operational parameters like:
temperature difference, voltage, ampere, watt will be study to know the thermal conductivity of copper
rod.
3. TABLE CONTAINS
• Certificate (2)
• Acknowledgement (3)
• Abstract (4)
• Table contents (5)
• List of figures (7)
• Chapter Topics Page No
• 1. Introduction. (9)
• 1.1.1 Heat transfer (9)
• 1.1.2 Conduction (9)
• 1.1.3 Convection (9)
• 1.1.4 Radiation. (10)
• 1.1.5 Thermal conductivity (11)
• 2. Literature review (12)
• 2.1.1 Thermal conductivities of different types of metals. (12)
• 2.1.2 Parameter affecting thermal conductivity (15)
• 3. Experimental Set Up (16)
• 3.1.1 Block Diagram (16)
• 3.1.2 Experimental Procedure (17)
• 3.1.3 Specification (18)
• 3.1.4 Components Used (19)
• 3.1.5 Copper Rod. (20)
• 3.1.6 M.s cylinder. (21)
• 3.1.7 Temperature Meter. (22)
• 3.1.8 Temperature Sensor. (23)
• 3.1.9 Nichrome wire heater. (25)
• 3.1.10 Digital Volt Meter (27)
• 3.1.11 Digital Ampere Meter (28)
• 3.1.12 Dimmerstat. (29)
• 3.1.13 Application (30)
4. List of figures
• No. Figures Page no.
• 1. Copper Rod (18)
• 2. M.s Cylinder. (19)
• 3. Temperature Meter (20)
• 4. Temperature sensor (21)
• 5. Nicrome wire Heater (23)
• 6. Digital volt Meter (24)
• 7. Digital Ampere Meter (25)
• 8. Dimmerstat (variac) (26)
• 9. Block diagram (27)
5. LIST OF TABLES
• NO. TOPIC PAGE NO
• 1. Thermal conductivity of different metals. (12).
• 2. Different material of wire heater. (22).
6. CHAPTER 1
INTRODUCTION
• 1.1.1 Introduction to heat transfer?
• Heat transfer ,also referred to simply as heat, is the movement of thermal energy from one thing to
another thing of different temperature. there are three diffrent ways the heat can transfer conduction,
convection and radiation.
7. 1.1.2 Conduction
• On a microscopic scale, heat conduction occurs as hot, rapidly moving or vibrating atoms and molecules
interact with neighboring atoms and molecules, transferring some of their energy (heat) to these
neighboring particles. In other words, heat is transferred by conduction when adjacent atoms vibrate
against one another, or as electrons move from one atom to another. Conduction is the most significant
means of heat transfer within a solid or between solid objects in thermal contect Fluids—especially
gases—are less conductive. Thermal contact conductance is the study of heat conduction between solid
bodies in contact. The process of heat transfer from one place to another place without the movement of
particles is called conduction. Example: Heat transfer through Metal rods. Steady state conduction is a
form of conduction that happens when the temperature difference driving the conduction is constant, so
that after an equilibration time, the spatial distribution of temperatures in the conducting object does not
change any further. In steady state conduction, the amount of heat entering a section is equal to amount
of heat coming out.
• Transient conduction occurs when the temperature within an object changes as a function of time.
Analysis of transient systems is more complex and often calls for the application of approximation theories
or numerical analysis by computer.
8. 1.1.3 Convection
• The flow of fluid may be forced by external processes, or sometimes (in gravitational fields) by buoyancy
forces caused when thermal energy expands the fluid (for example in a fire plume), thus influencing its
own transfer. The latter process is often called "natural convection". All convective processes also move
heat partly by diffusion, as well. Another form of convection is forced convection. In this case the fluid is
forced to flow by using a pump, fan or other mechanical means.
• Convective heat transfer, or convection, is the transfer of heat from one place to another by the
movement of fluids, a process that is essentially the transfer of heat via mass transfer. Bulk motion of fluid
enhances heat transfer in many physical situations, such as (for example) between a solid surface and the
fluid.[10] Convection is usually the dominant form of heat transfer in liquids and gases. Although sometimes
discussed as a third method of heat transfer, convection is usually used to describe the combined effects
of heat conduction within the fluid (diffusion) and heat transference by bulk fluid flow streaming.[11] The
process of transport by fluid streaming is known as advection, but pure advection is a term that is
generally associated only with mass transport in fluids, such as advection of pebbles in a river. In the case
of heat transfer in fluids, where transport by advection in a fluid is always also accompanied by transport
via heat diffusion (also known as heat conduction) the process of heat convection is understood to refer to
the sum of heat transport by advection and diffusion/conduction.
9. 1.1.4 Radiation.
• Thermal radiation occurs through a vacuum or any transparent medium (solid or fluid). It is the transfer of
energy by means of photons in electromagnetic waves governed by the same laws.
• Thermal radiation is energy emitted by matter as electromagnetic waves, due to the pool of thermal
energy in all matter with a temperature above absolute zero. Thermal radiation propagates without the
presence of matter through the vacuum of space.
• Thermal radiation is a direct result of the random movements of atoms and molecules in matter. Since
these atoms and molecules are composed of charged particles (protons and electrons), their movement
results in the emission of electromagnetic radiation, which carries energy away from the surface.
• The Stefan-Boltzmann equation, which describes the rate of transfer of radiant energy, is as follows for an
object in a vacuum:
10. 1.1.5 Introduction of thermal conductivity.
• The rate at which heat passes through a specified material, expressed as the amount of heat that flows
per unit time through a unit area with a temperature gradient of one degree per unit distance noun:
thermal conductivity.
• Is the physical property of the material denoting the ease with particular substances can accomplish the
transmission of thermal energy by molecular motion? Thermal conductivity of material is found to depend
on the chemical composition of the substances of which it is composed, the phase (i.e. Gas, Liquid or solid)
in which it exists, its crystalline structure if a solid, the temperature and pressure to which it is subjected,
and whether or not it is homogeneous material. The atoms in the rod that are exposed to the heat, gain
energy in the form of heat and transfer this energy to their neighbors, which then transfer the heat energy
to their neighboring atoms. In this manner, the energy is passed along through the length of the rod.
•
11. CHAPTER 2
Literature review
2.1.1 Thermal conductivities of different types of metals
Metal
Temperature
- t -
(oF)
Thermal
Conductivity
- k -
(Btu/(hr oF ft))
Temperature
- t -
(oC)
Thermal Conductivity
- k -
(W/m K)
Admiralty Brass 68 64 20 111
Aluminum, pure 68 118 20 204
200 124 93 215
400 144 204 249
Aluminum Bronze 68 44 20 76
Antimony 68 10.7 20 19
Beryllium 68 126 20 218
Beryllium Copper 68 38 20 66
Bismuth 68 4.9 20 8.5
18. 2.1.2 Parameter affecting thermal conductivity
• Here are the factors that affect the rate of conduction: Temperature difference. The greater the difference
in temperature between the two ends of the bar, the greater the rate of thermal energy transfer, so more
heat is transferred.
20. 3.1.2Experimental Procedure
• (1) First of all connect cold water supply at inlet of the cooling chamber of setup.
• (2) Make sure that drain valve is open.
• (3) Start water supply (1LPM) at constant water supply rate.
• (4) Now switch on the supply and make sure the zero position of all digital devices before switching on it.
• (5) Fix the power input to the heater with the help of Variac, Voltmeter and ammeter provided.
• (6) After 30 Minutes start recording the temperature of various point at each 5 minutes interval.
• (7) If temperature reading are same for three times, assume that state is achieved.
• (8) Record the final temperatures.
• (9) Repeat the above procedure for two different temperatures.
•
• CLOSING PROCEDURE:-
• (1) When experiment is over, switch off heater first.
• (2) Adjust Variac at zero.
• (3) Switch off the panel with the help of mains on/off switch given on the panel.
• (4) Switch off power supply to panel.
• (5) stop cold water supply.
21. 3.1.3SPECIFICATIONS
• Length of the metal Bar : 450mm
• Diameter of the metal bar : 25mm
• Test length of the bar : 235mm
• Total no of temperature sensors in the setup : 8 Nos.
• Number of Temperature sensors mounted on bar : 6 Nos.
• Number of Temperature sensors mounted on water jacket : 2 Nos.
• Type of Temperature sensors : RTD PT-100
• Heater : Nichrome
• Cooling jacket dia : 100mm
• Length of cooling jacket : 75mm
• Dimmer stat for heater coil : 2 Amp
• 230VAC Digital Voltmeter : 0 to 250 volt
• Digital Ammeter : 0 to 2.5
• Amps.
• Temperature indicator : 0 to 99.9 C
22. 3.1.4Components Used:
• 1..Metal rod
• 2. M.s cylinder
• 3. Temperature Meter
• 4. Temperature Sensor
• 5. heater
• 6. pump
• 7. flange
23. EXPLAINATION ABOUT COMPONENTS:
3.1.5 Copper Rod
• Copper is a chemical element with symbol CU and atomic number 29.it is a soft, malleable, and ductile
metal with very high thermal and electrical conductivity. A freshly exposed surface of pure copper has a
reddish-orange color. Copper is used as a conductor of heat and electricity, as a building material. And as a
constituent of various metal alloys, such as sterling silver used in jewelry, cupronickel used to make marine
hardware and coins, and constantan used in strain gauge and thermocouples for temperature
measurement.
• In this project we are using copper rod. Copper rod diameter is 25mm and length of copper bar is 425mm.
Test length of the copper bar is 200mm respectively. Copper rod. Copper rod diameter is 25mm and
length of copper bar is 425mm. Test length of the copper bar is 200mm respectively. Copper thermal
conductivity is the 9.2×10^-2 KCAL and 380 J respectively.
• Copper is a good conductor of heat. This means that if you heat one end of a piece of copper, the other
end will quickly reach the same temperature. Most metals are pretty good conductors; however, apart
from silver, copper is the best.
It is used in many heating applications because it doesn't corrode and has a high melting point. The only
other material that has similar resistance to corrosion is stainless steel. However, its thermal conductivity
is 30 times worse than that of copper.
24.
25. 3.1.6 M.s Cylinder
• Mild steel is made up to 5mm thick plate and it is passed through the bending machine and the cylinder is
made from it. Mild steel diameter is 100mm while the length of the mild steel cylinder is 250mm. On the
back end of the cylinder, the cover is attached as a flange. This flange is attached to the cylinder with the
help of nut and bolt. Copper rod is kept in the inner part of the cylinder. The cylinder is supported.
Insulation is done with copper rod in the cylinder.
• A water jacket is provided on one end of the cylinder so that one side of the copper rod is cold. Cylinder is
the main part of this project.
26.
27.
28. 3.1.7Temperature Meter
• What temperature should we measure from copper rod to temperature meter? In some part of full length
of copper rod, how much temperature can be easily measured. Temperature can be measured from
temperature range from T1 to T12. In the back part of the temperature meter, there are 12 ports, which
are given in connection with the temperature sensor in the 12 ports. The temperature sensor delivers
measured measurements digitally. Temperature meter gets in different range.`
29.
30. 3.1.8Temperature sensor
• A thermocouple is an electrical device consisting of two dissimilar electrical conductors forming electrical
junction at differing temperature. A thermocouple produces a temperature dependent voltage as a result
of the thermoelectric effect, and this voltage can be interpreted to measure temperature. Thermocouple
are a widely used type of temperature sensor.
• Commercial thermocouple are inexpensive interchangeable, are supplied with standard connectors, and
can measure a wide range of temperatures. In contrast to most other methods of temperature
measurement, thermocouple is self powered and require no external form of excitation. The main
limitation with thermocouples is accuracy; system errors of less the one degree Celsius can be difficult to
achieve.
• In 1821, the German physicist Thomas Thomas johann seebeck discovered that when different metals are
joined at the ends and there is a temperature difference between the joints, a magnetic field is observed.
At the time Seebeck referred to this as thermo-magnetism. The magnetic field he observed was later
shown to be due to thermo-electric current. In practical use, the voltage generated at a single junction of
two different types of wire is what is of interest as this can be used to measure temperature at very high
and low temperatures. The magnitude of the voltage depends on the types of wire used. Generally, the
voltage is in the microvolt range and care must be taken to obtain a usable measurement. Although very
little current flows, power can be generated by a single thermocouple junction. Power generation using
multiple thermocouples, as in a thermopile, is common.
31.
32. 3.1.9 Nichrome wire heater.
• Nichrome, a non-magnetic 80/20 alloy of nickel and chromium, is the most common resistance wire for
heating purposes because it has a high resistivity and resistance to oxidation at high temperatures. When
used as a heating element, resistance wire is usually wound into coils. One difficulty in using nichrome
wire is that common tin-based electrical solder will not bond with it, so the connections to the electrical
power must be made using other methods such as crimp connectors or screw terminals.
• Resistance wire is wire intended for making electrical resistors (which are used to control the amount of
current in a copper rod). It is better if the alloy used has a high resistivity, since a shorter wire can then be
used. In many situations, the stability of the resistor is of primary importance, and thus the
alloy's temperature coefficient of resistivity and corrosion resistance play a large part in material selection.
36. 3.1.10Digital volt Meter
• A voltmeter is an instrument used for measuring electrical potential difference between two points in an
electric circuit. Analog voltmeters move a pointer across a scale in proportion to the voltage of the circuit;
digital voltmeters give a numerical display of voltage by use of an analog to digital converter.
• A digital voltmeter (DVM) measures an unknown input voltage by converting the voltage to a digital value
and then displays the voltage in numeric form. DVMs are usually designed around a special type of analog-
to-digital converter called an integrating converter.
• DVM measurement accuracy is affected by many factors, including temperature, input impedance, and
DVM power supply voltage variations. Less expensive DVMs often have input resistance on the order of 10
MΩ. Precision DVMs can have input resistances of 1 GΩ or higher for the lower voltage ranges (e.g. less
than 20 V). To ensure that a DVM's accuracy is within the manufacturer's specified tolerances, it must be
periodically calibrated against a voltage standard such as the Weston cell.
37.
38. 3.1.11 Digital Ampere Meter
• An ammeter (from Ampere Meter) is a measuring instrument used to measure the current in a copper rod.
Electric currents are measured in amperes (A), hence the name. Instruments used to measure smaller
currents, in the milliampere or microampere range, are designated as milliammeters or micrometers.
• In much the same way as the analogue ammeter formed the basis for a wide variety of derived meters,
including voltmeters, the basic mechanism for a digital meter is a digital voltmeter mechanism, and other
types of meter are built around this.
• Digital ammeter designs use a shunt resistor to produce a calibrated voltage proportional to the current
flowing. This voltage is then measured by a digital voltmeter, through use of an analog to digital
converter (ADC); the digital display is calibrated to display the current through the shunt. Such instruments
are often calibrated to indicate the RMS value for a sine wave only, but many designs will indicate true
RMS within limitations of the wave crest factor
39.
40. 3.1.12 . Dimmerstat (Variac)
• Principal: - Variac is the trademark name of a variable autotransformer. Variac provides a voltage-
adjustable source of alternating current (AC) electricity. A variable autotransformer is a single-coil
transformer in which two portions of the same coil are used as the primary and the secondary.
• An autotransformer is an electrical transformer with only one winding. The "auto" prefix refers to the
single coil acting alone and not to any kind of automatic mechanism. In an autotransformer, portions of
the same winding act as both the primary and secondary side of the transformer. In contrast, an ordinary
transformer has separate primary and secondary winding which are not electrically connected.
• Autotransformers are frequently used in power applications to interconnect systems operating at different
voltage classes, for example 132 kV to 66 kV for transmission. Another application in industry is to adapt
machinery built (for example) for 480 V supplies to operate on a 600 V supply. They are also often used for
providing conversions between the two common domestic mains voltage bands in the world (100 V—
130 V and 200 V—250 V). The links between the UK 400 kV and 275 kV 'Super Grid' networks are normally
three phase autotransformers with taps at the common neutral end.
41.
42. 3.1.13APPLICATION OF THERMAL CONDUCTIVITY OF METAL ROD
• A) To measure the temperature gradient along the length of the metal (copper) rod.
• B) To determine the co-efficient of thermal conductivity of the metal (copper). ), ∆T/ ∆X is the temperature
gradient in the direction of heat flow