In this book, we solve the partial differential equation of the heat equation by first transforming it into an integral equation. We use exponential temperature profiles which satisfy the boundary conditions and also the initial condition. We also look at cases where ther is natural convection and go ahead and solve for both the transient and steady state solution and compare the mathematical findings with those got from experiment. We also go ahead an solve the heat equation in cylindrical coordinates. We explain alot of phenomena observed experimentally for example the melting of wax on the sides of a metal rod when heat is applied on one end
ANALTICAL SOLUTIONS TO THE HEAT EQUATION USING THE INTEGRAL METHODS.pdfWasswaderrick3
In this book, we solve the heat equation partial differential equation by first transforming it into an integral equation. We use exponential temperature profiles which satisfy the boundary conditions and also the initial condition. We also look at cases where ther is natural convection and go ahead and solve for both the transient and steady state solution. We also go ahead an solve the heat equation in cylindrical coordinates. We explain alot of phenomena observed experimentally for example the melting of wax on the sides of a metal rod when heat is applied on one end. For updated information about heat flow, follow the link below:
https://www.slideshare.net/Wasswaderrick3/analytic-solutions-to-the-heat-equation-using-integral-methods-with-experimental-resultspdf
Integral methods for the analytic solutions to the heat equation.pdfWasswaderrick3
In this book, we solve the heat equation partial differential equation by first transforming it into an integral equation. We use exponential temperature profiles which satisfy the boundary conditions and also the initial condition. We also look at cases where ther is natural convection and go ahead and solve for both the transient and steady state solution. We also go ahead an solve the heat equation in cylindrical coordinates. We explain alot of phenomena observed experimentally for example the melting of wax on the sides of a metal rod when heat is applied on one end. For updated information about heat flow, follow the link below:
https://www.slideshare.net/Wasswaderrick3/analytic-solutions-to-the-heat-equation-using-integral-methods-with-experimental-resultspdf
This document discusses analytical solutions to the heat equation using an integral method for various scenarios involving heat flow through semi-infinite and finite metal rods and cylinders. It first presents the analytical solution for a semi-infinite wall and provides an alternative derivation. It then explains how the Fourier law of heat conduction applies in steady state for a semi-infinite rod. Subsequent sections generalize the solutions to scenarios involving rods of different materials arranged in series, natural convection, different boundary conditions, finite lengths, zero flux boundaries, cylindrical coordinates, and temperature-dependent initial conditions.
In this book, we look at the analytical integral approach used to solve the heat equation. We look at different cases of boundary and initial conditions and we solve the heat equation using exponential temperature profiles that satisfy the boundary and initial conditions. We go ahead to look at predictions made by the solutions we have calculated and verify them experimentally. We look at different geometries including rectangular coordinates, cylindrical cordinates and solve their governing equations analytically. We look at the predictions of the transient state made by the solutions and verify them experimentally. We look at scenarios of semi-infinite rods, finite rods including semi-infinite cylinders and finite cylinders. We go ahead to develop the governing equation for heat loss by convection for a liquid in a container.In all the above solutions, we used the integral approach to solve for the solutions. We compare the Fourier series solution to our solution and we realise that the Fourier solution is approximate since it involves summing terms to infinity yet we notice that our solution is exact. We look at cases where theres is both conduction and natural convection at the sides of the rod and solve the governing equations for given boundary and initial conditions. We realise that our method of approach can be used to solve the heat equation for any type of boundary conditions.
TRANSIENT AND STEADY STATE HEAT CONDUCTION WITH NO LATERAL CONVECTION SOLVED ...Wasswaderrick3
In this book we go ahead and solve for the transient and steady state heat conduction phenomena in one dimensional heat flow with no lateral convection. It is known that there exist a Fourier series method but the problem of this method is that it is an approximate method since it involves summing up to infinite number of terms which we can never achieve in practice without approximating. In this book we develop an analytic solution to the heat equation with no lateral convection by using the already derived hyperbolic temperature profile functions in literature and solve the heat equation using these functions and the integral equation method and get a solution of the time dependent parameter δ which we substitute in the temperature profile. We deal with different types of boundary conditions and get their solutions. In solving for the steady state temperatures, we use the L’hopital’s rule since we get undefined limits when we substitute for time tending to infinity in some cases. We realize that the steady state temperature profile agrees with theory.
SEMI-INFINITE ROD SOLUTION FOR TRANSIENT AND STEADY STATE.pdfWasswaderrick3
For the case of conduction of semi-infinite metal rod in natural convection, we postulate that the temperature profile which satisfies the boundary conditions and the initial condition is the exponential temperature profile. We go ahead and solve the heat equation using this temperature profile and the integral approach and the solution got is used to explain what is observed in the transient and steady state. We notice that the prediction made by the theory is not exactly what is observed with an intercept term which comes in. To account for this intercept, we postulate that there’s convection at the hot end. This accounts for the observed intercept. This analysis can be extended to metal rods of finite length with given boundary conditions and different geometries.
RADIAL HEAT CONDUCTION SOLVED USING THE INTEGRAL EQUATION .pdfWasswaderrick3
We look at the case of radial heat flow. Again, in radial heat flow, the temperature profiles that satisfy the boundary and initial conditions are the exponential and hyperbolic functions as derived in literature of conduction in fins. We use the technique of transforming the PDE into an integral equation. But in the case of radial heat flow, we have to multiply through by r the heat equation and then introduce integrals. We do this to avoid introducing integrals of the form of the exponential integral whose solutions cannot be expressed in the form of a simple mathematical function. We look at the case of a semi-infinite hollow cylinder for both insulated and non-insulated cases and then find the solution. We also look at cases of finite radius hollow cylinders subject to given boundary conditions. We notice that the solutions got for finite radius hollow cylinders do not reduce to those of semi-infinite hollow cylinders. We conclude by saying that this same analysis can be extended to spherical co-ordinates heat conduction.
GENERAL MATHEMATICAL THEORY OF HEAT CONDUCTION USING THE INTEGRAL HEAT EQUATI...Wasswaderrick3
In this book, we look at using the integral heat equation as the general method of solving the heat equation subject to given boundary conditions. We begin first by looking at x- directional heat conduction and look at the case of the insulated metal rod first. It is known from literature that the Fourier series yield a solution to this problem for given boundary conditions. But on analyzing the solution got, we notice that it is made up of an infinite number of terms and what this means is that we shall only have an approximate solution since we can’t in practice add up all the terms to infinity. To solve this problem, we solved the heat equation by first transforming it into an integral equation and then find an exact solution as shall be shown in the text later. In solving the heat equation, the temperature profiles that satisfy the heat equation are exponential temperature profiles and hyperbolic temperature profiles as derived in literature for heat conduction in fins. For this case of insulate metal rod, we invoke L’hopital’s rule to get the steady state temperature profile. We then extend this integral equation approach to the case where there is lateral convection along the metal rod and get also both the transient and steady state solution which agrees with theory for steady state heat conduction.
After that, we look at the case of radial heat flow. Again, in radial heat flow, the temperature profiles that satisfy the boundary and initial conditions are the exponential and hyperbolic functions as derived in literature of conduction in fins. We use the same technique of transforming the PDE into an integral equation. But in the case of radial heat flow, we have to multiply through by r the heat equation and then introduce integrals. We do this to avoid introducing integrals of the form of the exponential integral whose solutions cannot be expressed in the form of a simple mathematical function. We look at the case of a semi-infinite hollow cylinder for both insulated and non-insulated cases and then find the solution. We also look at cases of finite radius hollow cylinders subject to given boundary conditions. We notice that the solutions got for finite radius hollow cylinders do not reduce to those of semi-infinite hollow cylinders as was the case for x-directional heat flow. We conclude by saying that this same analysis can be extended to spherical co-ordinates heat conduction
ANALTICAL SOLUTIONS TO THE HEAT EQUATION USING THE INTEGRAL METHODS.pdfWasswaderrick3
In this book, we solve the heat equation partial differential equation by first transforming it into an integral equation. We use exponential temperature profiles which satisfy the boundary conditions and also the initial condition. We also look at cases where ther is natural convection and go ahead and solve for both the transient and steady state solution. We also go ahead an solve the heat equation in cylindrical coordinates. We explain alot of phenomena observed experimentally for example the melting of wax on the sides of a metal rod when heat is applied on one end. For updated information about heat flow, follow the link below:
https://www.slideshare.net/Wasswaderrick3/analytic-solutions-to-the-heat-equation-using-integral-methods-with-experimental-resultspdf
Integral methods for the analytic solutions to the heat equation.pdfWasswaderrick3
In this book, we solve the heat equation partial differential equation by first transforming it into an integral equation. We use exponential temperature profiles which satisfy the boundary conditions and also the initial condition. We also look at cases where ther is natural convection and go ahead and solve for both the transient and steady state solution. We also go ahead an solve the heat equation in cylindrical coordinates. We explain alot of phenomena observed experimentally for example the melting of wax on the sides of a metal rod when heat is applied on one end. For updated information about heat flow, follow the link below:
https://www.slideshare.net/Wasswaderrick3/analytic-solutions-to-the-heat-equation-using-integral-methods-with-experimental-resultspdf
This document discusses analytical solutions to the heat equation using an integral method for various scenarios involving heat flow through semi-infinite and finite metal rods and cylinders. It first presents the analytical solution for a semi-infinite wall and provides an alternative derivation. It then explains how the Fourier law of heat conduction applies in steady state for a semi-infinite rod. Subsequent sections generalize the solutions to scenarios involving rods of different materials arranged in series, natural convection, different boundary conditions, finite lengths, zero flux boundaries, cylindrical coordinates, and temperature-dependent initial conditions.
In this book, we look at the analytical integral approach used to solve the heat equation. We look at different cases of boundary and initial conditions and we solve the heat equation using exponential temperature profiles that satisfy the boundary and initial conditions. We go ahead to look at predictions made by the solutions we have calculated and verify them experimentally. We look at different geometries including rectangular coordinates, cylindrical cordinates and solve their governing equations analytically. We look at the predictions of the transient state made by the solutions and verify them experimentally. We look at scenarios of semi-infinite rods, finite rods including semi-infinite cylinders and finite cylinders. We go ahead to develop the governing equation for heat loss by convection for a liquid in a container.In all the above solutions, we used the integral approach to solve for the solutions. We compare the Fourier series solution to our solution and we realise that the Fourier solution is approximate since it involves summing terms to infinity yet we notice that our solution is exact. We look at cases where theres is both conduction and natural convection at the sides of the rod and solve the governing equations for given boundary and initial conditions. We realise that our method of approach can be used to solve the heat equation for any type of boundary conditions.
TRANSIENT AND STEADY STATE HEAT CONDUCTION WITH NO LATERAL CONVECTION SOLVED ...Wasswaderrick3
In this book we go ahead and solve for the transient and steady state heat conduction phenomena in one dimensional heat flow with no lateral convection. It is known that there exist a Fourier series method but the problem of this method is that it is an approximate method since it involves summing up to infinite number of terms which we can never achieve in practice without approximating. In this book we develop an analytic solution to the heat equation with no lateral convection by using the already derived hyperbolic temperature profile functions in literature and solve the heat equation using these functions and the integral equation method and get a solution of the time dependent parameter δ which we substitute in the temperature profile. We deal with different types of boundary conditions and get their solutions. In solving for the steady state temperatures, we use the L’hopital’s rule since we get undefined limits when we substitute for time tending to infinity in some cases. We realize that the steady state temperature profile agrees with theory.
SEMI-INFINITE ROD SOLUTION FOR TRANSIENT AND STEADY STATE.pdfWasswaderrick3
For the case of conduction of semi-infinite metal rod in natural convection, we postulate that the temperature profile which satisfies the boundary conditions and the initial condition is the exponential temperature profile. We go ahead and solve the heat equation using this temperature profile and the integral approach and the solution got is used to explain what is observed in the transient and steady state. We notice that the prediction made by the theory is not exactly what is observed with an intercept term which comes in. To account for this intercept, we postulate that there’s convection at the hot end. This accounts for the observed intercept. This analysis can be extended to metal rods of finite length with given boundary conditions and different geometries.
RADIAL HEAT CONDUCTION SOLVED USING THE INTEGRAL EQUATION .pdfWasswaderrick3
We look at the case of radial heat flow. Again, in radial heat flow, the temperature profiles that satisfy the boundary and initial conditions are the exponential and hyperbolic functions as derived in literature of conduction in fins. We use the technique of transforming the PDE into an integral equation. But in the case of radial heat flow, we have to multiply through by r the heat equation and then introduce integrals. We do this to avoid introducing integrals of the form of the exponential integral whose solutions cannot be expressed in the form of a simple mathematical function. We look at the case of a semi-infinite hollow cylinder for both insulated and non-insulated cases and then find the solution. We also look at cases of finite radius hollow cylinders subject to given boundary conditions. We notice that the solutions got for finite radius hollow cylinders do not reduce to those of semi-infinite hollow cylinders. We conclude by saying that this same analysis can be extended to spherical co-ordinates heat conduction.
GENERAL MATHEMATICAL THEORY OF HEAT CONDUCTION USING THE INTEGRAL HEAT EQUATI...Wasswaderrick3
In this book, we look at using the integral heat equation as the general method of solving the heat equation subject to given boundary conditions. We begin first by looking at x- directional heat conduction and look at the case of the insulated metal rod first. It is known from literature that the Fourier series yield a solution to this problem for given boundary conditions. But on analyzing the solution got, we notice that it is made up of an infinite number of terms and what this means is that we shall only have an approximate solution since we can’t in practice add up all the terms to infinity. To solve this problem, we solved the heat equation by first transforming it into an integral equation and then find an exact solution as shall be shown in the text later. In solving the heat equation, the temperature profiles that satisfy the heat equation are exponential temperature profiles and hyperbolic temperature profiles as derived in literature for heat conduction in fins. For this case of insulate metal rod, we invoke L’hopital’s rule to get the steady state temperature profile. We then extend this integral equation approach to the case where there is lateral convection along the metal rod and get also both the transient and steady state solution which agrees with theory for steady state heat conduction.
After that, we look at the case of radial heat flow. Again, in radial heat flow, the temperature profiles that satisfy the boundary and initial conditions are the exponential and hyperbolic functions as derived in literature of conduction in fins. We use the same technique of transforming the PDE into an integral equation. But in the case of radial heat flow, we have to multiply through by r the heat equation and then introduce integrals. We do this to avoid introducing integrals of the form of the exponential integral whose solutions cannot be expressed in the form of a simple mathematical function. We look at the case of a semi-infinite hollow cylinder for both insulated and non-insulated cases and then find the solution. We also look at cases of finite radius hollow cylinders subject to given boundary conditions. We notice that the solutions got for finite radius hollow cylinders do not reduce to those of semi-infinite hollow cylinders as was the case for x-directional heat flow. We conclude by saying that this same analysis can be extended to spherical co-ordinates heat conduction
DERIVATION OF MODIFIED BERNOULLI EQUATION WITH VISCOUS EFFECTS AND TERMINAL V...Wasswaderrick3
In this book, we use conservation of energy techniques on a fluid element to derive the Modified Bernoulli equation of flow with viscous or friction effects. We derive the general equation of flow/ velocity and then from this we derive the Pouiselle flow equation, the transition flow equation and the turbulent flow equation. In the situations where there are no viscous effects , the equation reduces to the Bernoulli equation. From experimental results, we are able to include other terms in the Bernoulli equation. We also look at cases where pressure gradients exist. We use the Modified Bernoulli equation to derive equations of flow rate for pipes of different cross sectional areas connected together. We also extend our techniques of energy conservation to a sphere falling in a viscous medium under the effect of gravity. We demonstrate Stokes equation of terminal velocity and turbulent flow equation. We look at a way of calculating the time taken for a body to fall in a viscous medium. We also look at the general equation of terminal velocity.
FUNDAMENTALS OF FLUID FLOW 3rd Edition .pdfWasswaderrick3
In this book we use energy conservation techniques on a differential element and derive the governing equation from which we get the Toricelli , Pouiselle , transition and turbulent flow equations all from one equation. The most important value in using the energy conservation techniques is to first get the Friction factor by solving the corresponding Navier Stoke's equation for a given geometry of pipe for laminar flow and then find the Friction factor whose value we use to derive the Toricelli governing equation, the laminar flow equation, turbulent flow and transition flow equations. We note that in Toricelli flow the length of the pipe is reduced to zero and we go ahead to derive the governing equation. We go ahead to look at the head loss equations and derive the friction factors from them. we also go ahead to look at a spherical body falling under the influence of gravity alone and we derive the governing equations and terminal velocity equations. Other phenomena are explained too
n this book we use energy conservation techniques on a differential element and derive the governing equation from which we get the Toricelli , Pouiselle , transition and turbulent flow equations all from one equation. The most important value in using the energy conservation techniques is to first get the Friction factor by solving the corresponding Navier Stoke's equation for a given geometry of pipe for laminar flow and then find the Friction factor whose value we use to derive the Toricelli governing equation, the laminar flow equation, turbulent flow and transition flow equations. We note that in Toricelli flow the length of the pipe is reduced to zero and we go ahead to derive the governing equation. We go ahead to look at the head loss equations and derive the friction factors from them. we also go ahead to look at a spherical body falling under the influence of gravity alone and we derive the governing equations and terminal velocity equations. Other phenomena are explained too
FUNDAMENTALS OF FLUID FLOW 3rd Edition.pdfWasswaderrick3
In this book we use energy conservation techniques on a differential element and derive the governing equation from which we get the Toricelli , Pouiselle , transition and turbulent flow equations all from one equation. The most important value in using the energy conservation techniques is to first get the Friction factor by solving the corresponding Navier Stoke's equation for a given geometry of pipe for laminar flow and then find the Friction factor whose value we use to derive the Toricelli governing equation, the laminar flow equation, turbulent flow and transition flow equations. We note that in Toricelli flow the length of the pipe is reduced to zero and we go ahead to derive the governing equation. We go ahead to look at the head loss equations and derive the friction factors from them. we also go ahead to look at a spherical body falling under the influence of gravity alone and we derive the governing equations and terminal velocity equations. Other phenomena are explained too
THE TEMPERATURE PROFILE SOLUTION IN NATURAL CONVECTION SOLVED MATHEMATICALLY.pdfWasswaderrick3
In this book we derive the generalized cooling law in natural convection for all temperature excesses from empirical results using the fact that Newton's law of cooling is obeyed for temperature excesses less than 30C. We derive an equation from empirical results that predicts the rate of cooling for a given temperature excess and then integrate and solve this equation to give us a straight line that predicts the temperature at a given time for a cooling body in natural convection for all temperature excesses
THE COOLING CURVE SOLUTION IN NATURAL CONVECTION EXPLAINED.pdfWasswaderrick3
In this book we derive the generalized cooling law in natural convection for all temperature excesses from empirical results using the fact that Newton's law of cooling is obeyed for temperature excesses less than 30C. We derive an equation from empirical results that predicts the rate of cooling for a given temperature excess and then integrate and solve this equation to give us a straight line that predicts the temperature at a given time for a cooling body in natural convection for all temperature excesses
THE RATE OF COOLING IN NATURAL CONVECTION EXPLAINED AND SOLVED.pdfWasswaderrick3
In this book we derive the generalized cooling law in natural convection for all temperature excesses from empirical results using the fact that Newton's law of cooling is obeyed for temperature excesses less than 30C. We derive an equation from empirical results that predicts the rate of cooling for a given temperature excess and then integrate and solve this equation to give us a straight line that predicts the temperature at a given time for a cooling body in natural convection for all temperature excesses
TRANSIENT STATE HEAT CONDUCTION SOLVED.pdfWasswaderrick3
Here are the key claims, findings, and conclusions from the text:
Claims:
- Transient state heat transfer in a cylindrical metal rod with one end heated can be modeled mathematically. - The temperature profile along the rod depends on whether the rod is semi-infinite or finite.
- There is convection at the heated end of the rod as investigated experimentally in addition to conduction along the rod.
Findings: - For a semi-infinite rod, the temperature profile decays exponentially with distance from the heated end.
- For a finite rod with no flux at the far end, the temperature profile involves hyperbolic cosine functions.
- For a finite rod with convection at the far end, the temperature profile also depends on hyperbolic sine functions. - The models match experimental measurements of melting wax particles along aluminum rods. We deduce from experiment that there’s convection at the hot end and that the heat transfer coefficient at the far end varies with length of the metal rod.
Conclusions:
- The integral transform approach allows deriving temperature profiles satisfying boundary conditions.
- There is convection at the hot end, and the heat transfer coefficient at the far end varies with length L as investigated experimentally.
- The simple models provide good agreement with experimental temperature profiles in aluminum rods. In summary, the text presents mathematical models of heat transfer in metal rods that agree with experiments and provide insight into the temperature profiles by considering different boundary conditions. The key findings relate to the functional forms of the temperature profiles in semi-infinite versus finite rods.
DERIVATION OF THE MODIFIED BERNOULLI EQUATION WITH VISCOUS EFFECTS AND TERMIN...Wasswaderrick3
In this book, we use conservation of energy techniques on a fluid element to derive the Modified Bernoulli equation of flow with viscous or friction effects. We derive the general equation of flow/ velocity and then from this we derive the Pouiselle flow equation, the transition flow equation and the turbulent flow equation. In the situations where there are no viscous effects , the equation reduces to the Bernoulli equation. From experimental results, we are able to include other terms in the Bernoulli equation. We also look at cases where pressure gradients exist. We use the Modified Bernoulli equation to derive equations of flow rate for pipes of different cross sectional areas connected together. We also extend our techniques of energy conservation to a sphere falling in a viscous medium under the effect of gravity. We demonstrate Stokes equation of terminal velocity and turbulent flow equation. We look at a way of calculating the time taken for a body to fall in a viscous medium. We also look at the general equation of terminal velocity.
DEEP LEARNING FOR SMART GRID INTRUSION DETECTION: A HYBRID CNN-LSTM-BASED MODELgerogepatton
As digital technology becomes more deeply embedded in power systems, protecting the communication
networks of Smart Grids (SG) has emerged as a critical concern. Distributed Network Protocol 3 (DNP3)
represents a multi-tiered application layer protocol extensively utilized in Supervisory Control and Data
Acquisition (SCADA)-based smart grids to facilitate real-time data gathering and control functionalities.
Robust Intrusion Detection Systems (IDS) are necessary for early threat detection and mitigation because
of the interconnection of these networks, which makes them vulnerable to a variety of cyberattacks. To
solve this issue, this paper develops a hybrid Deep Learning (DL) model specifically designed for intrusion
detection in smart grids. The proposed approach is a combination of the Convolutional Neural Network
(CNN) and the Long-Short-Term Memory algorithms (LSTM). We employed a recent intrusion detection
dataset (DNP3), which focuses on unauthorized commands and Denial of Service (DoS) cyberattacks, to
train and test our model. The results of our experiments show that our CNN-LSTM method is much better
at finding smart grid intrusions than other deep learning algorithms used for classification. In addition,
our proposed approach improves accuracy, precision, recall, and F1 score, achieving a high detection
accuracy rate of 99.50%.
Literature Review Basics and Understanding Reference Management.pptxDr Ramhari Poudyal
Three-day training on academic research focuses on analytical tools at United Technical College, supported by the University Grant Commission, Nepal. 24-26 May 2024
Batteries -Introduction – Types of Batteries – discharging and charging of battery - characteristics of battery –battery rating- various tests on battery- – Primary battery: silver button cell- Secondary battery :Ni-Cd battery-modern battery: lithium ion battery-maintenance of batteries-choices of batteries for electric vehicle applications.
Fuel Cells: Introduction- importance and classification of fuel cells - description, principle, components, applications of fuel cells: H2-O2 fuel cell, alkaline fuel cell, molten carbonate fuel cell and direct methanol fuel cells.
A review on techniques and modelling methodologies used for checking electrom...nooriasukmaningtyas
The proper function of the integrated circuit (IC) in an inhibiting electromagnetic environment has always been a serious concern throughout the decades of revolution in the world of electronics, from disjunct devices to today’s integrated circuit technology, where billions of transistors are combined on a single chip. The automotive industry and smart vehicles in particular, are confronting design issues such as being prone to electromagnetic interference (EMI). Electronic control devices calculate incorrect outputs because of EMI and sensors give misleading values which can prove fatal in case of automotives. In this paper, the authors have non exhaustively tried to review research work concerned with the investigation of EMI in ICs and prediction of this EMI using various modelling methodologies and measurement setups.
KuberTENes Birthday Bash Guadalajara - K8sGPT first impressionsVictor Morales
K8sGPT is a tool that analyzes and diagnoses Kubernetes clusters. This presentation was used to share the requirements and dependencies to deploy K8sGPT in a local environment.
A SYSTEMATIC RISK ASSESSMENT APPROACH FOR SECURING THE SMART IRRIGATION SYSTEMSIJNSA Journal
The smart irrigation system represents an innovative approach to optimize water usage in agricultural and landscaping practices. The integration of cutting-edge technologies, including sensors, actuators, and data analysis, empowers this system to provide accurate monitoring and control of irrigation processes by leveraging real-time environmental conditions. The main objective of a smart irrigation system is to optimize water efficiency, minimize expenses, and foster the adoption of sustainable water management methods. This paper conducts a systematic risk assessment by exploring the key components/assets and their functionalities in the smart irrigation system. The crucial role of sensors in gathering data on soil moisture, weather patterns, and plant well-being is emphasized in this system. These sensors enable intelligent decision-making in irrigation scheduling and water distribution, leading to enhanced water efficiency and sustainable water management practices. Actuators enable automated control of irrigation devices, ensuring precise and targeted water delivery to plants. Additionally, the paper addresses the potential threat and vulnerabilities associated with smart irrigation systems. It discusses limitations of the system, such as power constraints and computational capabilities, and calculates the potential security risks. The paper suggests possible risk treatment methods for effective secure system operation. In conclusion, the paper emphasizes the significant benefits of implementing smart irrigation systems, including improved water conservation, increased crop yield, and reduced environmental impact. Additionally, based on the security analysis conducted, the paper recommends the implementation of countermeasures and security approaches to address vulnerabilities and ensure the integrity and reliability of the system. By incorporating these measures, smart irrigation technology can revolutionize water management practices in agriculture, promoting sustainability, resource efficiency, and safeguarding against potential security threats.
Advanced control scheme of doubly fed induction generator for wind turbine us...IJECEIAES
This paper describes a speed control device for generating electrical energy on an electricity network based on the doubly fed induction generator (DFIG) used for wind power conversion systems. At first, a double-fed induction generator model was constructed. A control law is formulated to govern the flow of energy between the stator of a DFIG and the energy network using three types of controllers: proportional integral (PI), sliding mode controller (SMC) and second order sliding mode controller (SOSMC). Their different results in terms of power reference tracking, reaction to unexpected speed fluctuations, sensitivity to perturbations, and resilience against machine parameter alterations are compared. MATLAB/Simulink was used to conduct the simulations for the preceding study. Multiple simulations have shown very satisfying results, and the investigations demonstrate the efficacy and power-enhancing capabilities of the suggested control system.
DERIVATION OF MODIFIED BERNOULLI EQUATION WITH VISCOUS EFFECTS AND TERMINAL V...Wasswaderrick3
In this book, we use conservation of energy techniques on a fluid element to derive the Modified Bernoulli equation of flow with viscous or friction effects. We derive the general equation of flow/ velocity and then from this we derive the Pouiselle flow equation, the transition flow equation and the turbulent flow equation. In the situations where there are no viscous effects , the equation reduces to the Bernoulli equation. From experimental results, we are able to include other terms in the Bernoulli equation. We also look at cases where pressure gradients exist. We use the Modified Bernoulli equation to derive equations of flow rate for pipes of different cross sectional areas connected together. We also extend our techniques of energy conservation to a sphere falling in a viscous medium under the effect of gravity. We demonstrate Stokes equation of terminal velocity and turbulent flow equation. We look at a way of calculating the time taken for a body to fall in a viscous medium. We also look at the general equation of terminal velocity.
FUNDAMENTALS OF FLUID FLOW 3rd Edition .pdfWasswaderrick3
In this book we use energy conservation techniques on a differential element and derive the governing equation from which we get the Toricelli , Pouiselle , transition and turbulent flow equations all from one equation. The most important value in using the energy conservation techniques is to first get the Friction factor by solving the corresponding Navier Stoke's equation for a given geometry of pipe for laminar flow and then find the Friction factor whose value we use to derive the Toricelli governing equation, the laminar flow equation, turbulent flow and transition flow equations. We note that in Toricelli flow the length of the pipe is reduced to zero and we go ahead to derive the governing equation. We go ahead to look at the head loss equations and derive the friction factors from them. we also go ahead to look at a spherical body falling under the influence of gravity alone and we derive the governing equations and terminal velocity equations. Other phenomena are explained too
n this book we use energy conservation techniques on a differential element and derive the governing equation from which we get the Toricelli , Pouiselle , transition and turbulent flow equations all from one equation. The most important value in using the energy conservation techniques is to first get the Friction factor by solving the corresponding Navier Stoke's equation for a given geometry of pipe for laminar flow and then find the Friction factor whose value we use to derive the Toricelli governing equation, the laminar flow equation, turbulent flow and transition flow equations. We note that in Toricelli flow the length of the pipe is reduced to zero and we go ahead to derive the governing equation. We go ahead to look at the head loss equations and derive the friction factors from them. we also go ahead to look at a spherical body falling under the influence of gravity alone and we derive the governing equations and terminal velocity equations. Other phenomena are explained too
FUNDAMENTALS OF FLUID FLOW 3rd Edition.pdfWasswaderrick3
In this book we use energy conservation techniques on a differential element and derive the governing equation from which we get the Toricelli , Pouiselle , transition and turbulent flow equations all from one equation. The most important value in using the energy conservation techniques is to first get the Friction factor by solving the corresponding Navier Stoke's equation for a given geometry of pipe for laminar flow and then find the Friction factor whose value we use to derive the Toricelli governing equation, the laminar flow equation, turbulent flow and transition flow equations. We note that in Toricelli flow the length of the pipe is reduced to zero and we go ahead to derive the governing equation. We go ahead to look at the head loss equations and derive the friction factors from them. we also go ahead to look at a spherical body falling under the influence of gravity alone and we derive the governing equations and terminal velocity equations. Other phenomena are explained too
THE TEMPERATURE PROFILE SOLUTION IN NATURAL CONVECTION SOLVED MATHEMATICALLY.pdfWasswaderrick3
In this book we derive the generalized cooling law in natural convection for all temperature excesses from empirical results using the fact that Newton's law of cooling is obeyed for temperature excesses less than 30C. We derive an equation from empirical results that predicts the rate of cooling for a given temperature excess and then integrate and solve this equation to give us a straight line that predicts the temperature at a given time for a cooling body in natural convection for all temperature excesses
THE COOLING CURVE SOLUTION IN NATURAL CONVECTION EXPLAINED.pdfWasswaderrick3
In this book we derive the generalized cooling law in natural convection for all temperature excesses from empirical results using the fact that Newton's law of cooling is obeyed for temperature excesses less than 30C. We derive an equation from empirical results that predicts the rate of cooling for a given temperature excess and then integrate and solve this equation to give us a straight line that predicts the temperature at a given time for a cooling body in natural convection for all temperature excesses
THE RATE OF COOLING IN NATURAL CONVECTION EXPLAINED AND SOLVED.pdfWasswaderrick3
In this book we derive the generalized cooling law in natural convection for all temperature excesses from empirical results using the fact that Newton's law of cooling is obeyed for temperature excesses less than 30C. We derive an equation from empirical results that predicts the rate of cooling for a given temperature excess and then integrate and solve this equation to give us a straight line that predicts the temperature at a given time for a cooling body in natural convection for all temperature excesses
TRANSIENT STATE HEAT CONDUCTION SOLVED.pdfWasswaderrick3
Here are the key claims, findings, and conclusions from the text:
Claims:
- Transient state heat transfer in a cylindrical metal rod with one end heated can be modeled mathematically. - The temperature profile along the rod depends on whether the rod is semi-infinite or finite.
- There is convection at the heated end of the rod as investigated experimentally in addition to conduction along the rod.
Findings: - For a semi-infinite rod, the temperature profile decays exponentially with distance from the heated end.
- For a finite rod with no flux at the far end, the temperature profile involves hyperbolic cosine functions.
- For a finite rod with convection at the far end, the temperature profile also depends on hyperbolic sine functions. - The models match experimental measurements of melting wax particles along aluminum rods. We deduce from experiment that there’s convection at the hot end and that the heat transfer coefficient at the far end varies with length of the metal rod.
Conclusions:
- The integral transform approach allows deriving temperature profiles satisfying boundary conditions.
- There is convection at the hot end, and the heat transfer coefficient at the far end varies with length L as investigated experimentally.
- The simple models provide good agreement with experimental temperature profiles in aluminum rods. In summary, the text presents mathematical models of heat transfer in metal rods that agree with experiments and provide insight into the temperature profiles by considering different boundary conditions. The key findings relate to the functional forms of the temperature profiles in semi-infinite versus finite rods.
DERIVATION OF THE MODIFIED BERNOULLI EQUATION WITH VISCOUS EFFECTS AND TERMIN...Wasswaderrick3
In this book, we use conservation of energy techniques on a fluid element to derive the Modified Bernoulli equation of flow with viscous or friction effects. We derive the general equation of flow/ velocity and then from this we derive the Pouiselle flow equation, the transition flow equation and the turbulent flow equation. In the situations where there are no viscous effects , the equation reduces to the Bernoulli equation. From experimental results, we are able to include other terms in the Bernoulli equation. We also look at cases where pressure gradients exist. We use the Modified Bernoulli equation to derive equations of flow rate for pipes of different cross sectional areas connected together. We also extend our techniques of energy conservation to a sphere falling in a viscous medium under the effect of gravity. We demonstrate Stokes equation of terminal velocity and turbulent flow equation. We look at a way of calculating the time taken for a body to fall in a viscous medium. We also look at the general equation of terminal velocity.
DEEP LEARNING FOR SMART GRID INTRUSION DETECTION: A HYBRID CNN-LSTM-BASED MODELgerogepatton
As digital technology becomes more deeply embedded in power systems, protecting the communication
networks of Smart Grids (SG) has emerged as a critical concern. Distributed Network Protocol 3 (DNP3)
represents a multi-tiered application layer protocol extensively utilized in Supervisory Control and Data
Acquisition (SCADA)-based smart grids to facilitate real-time data gathering and control functionalities.
Robust Intrusion Detection Systems (IDS) are necessary for early threat detection and mitigation because
of the interconnection of these networks, which makes them vulnerable to a variety of cyberattacks. To
solve this issue, this paper develops a hybrid Deep Learning (DL) model specifically designed for intrusion
detection in smart grids. The proposed approach is a combination of the Convolutional Neural Network
(CNN) and the Long-Short-Term Memory algorithms (LSTM). We employed a recent intrusion detection
dataset (DNP3), which focuses on unauthorized commands and Denial of Service (DoS) cyberattacks, to
train and test our model. The results of our experiments show that our CNN-LSTM method is much better
at finding smart grid intrusions than other deep learning algorithms used for classification. In addition,
our proposed approach improves accuracy, precision, recall, and F1 score, achieving a high detection
accuracy rate of 99.50%.
Literature Review Basics and Understanding Reference Management.pptxDr Ramhari Poudyal
Three-day training on academic research focuses on analytical tools at United Technical College, supported by the University Grant Commission, Nepal. 24-26 May 2024
Batteries -Introduction – Types of Batteries – discharging and charging of battery - characteristics of battery –battery rating- various tests on battery- – Primary battery: silver button cell- Secondary battery :Ni-Cd battery-modern battery: lithium ion battery-maintenance of batteries-choices of batteries for electric vehicle applications.
Fuel Cells: Introduction- importance and classification of fuel cells - description, principle, components, applications of fuel cells: H2-O2 fuel cell, alkaline fuel cell, molten carbonate fuel cell and direct methanol fuel cells.
A review on techniques and modelling methodologies used for checking electrom...nooriasukmaningtyas
The proper function of the integrated circuit (IC) in an inhibiting electromagnetic environment has always been a serious concern throughout the decades of revolution in the world of electronics, from disjunct devices to today’s integrated circuit technology, where billions of transistors are combined on a single chip. The automotive industry and smart vehicles in particular, are confronting design issues such as being prone to electromagnetic interference (EMI). Electronic control devices calculate incorrect outputs because of EMI and sensors give misleading values which can prove fatal in case of automotives. In this paper, the authors have non exhaustively tried to review research work concerned with the investigation of EMI in ICs and prediction of this EMI using various modelling methodologies and measurement setups.
KuberTENes Birthday Bash Guadalajara - K8sGPT first impressionsVictor Morales
K8sGPT is a tool that analyzes and diagnoses Kubernetes clusters. This presentation was used to share the requirements and dependencies to deploy K8sGPT in a local environment.
A SYSTEMATIC RISK ASSESSMENT APPROACH FOR SECURING THE SMART IRRIGATION SYSTEMSIJNSA Journal
The smart irrigation system represents an innovative approach to optimize water usage in agricultural and landscaping practices. The integration of cutting-edge technologies, including sensors, actuators, and data analysis, empowers this system to provide accurate monitoring and control of irrigation processes by leveraging real-time environmental conditions. The main objective of a smart irrigation system is to optimize water efficiency, minimize expenses, and foster the adoption of sustainable water management methods. This paper conducts a systematic risk assessment by exploring the key components/assets and their functionalities in the smart irrigation system. The crucial role of sensors in gathering data on soil moisture, weather patterns, and plant well-being is emphasized in this system. These sensors enable intelligent decision-making in irrigation scheduling and water distribution, leading to enhanced water efficiency and sustainable water management practices. Actuators enable automated control of irrigation devices, ensuring precise and targeted water delivery to plants. Additionally, the paper addresses the potential threat and vulnerabilities associated with smart irrigation systems. It discusses limitations of the system, such as power constraints and computational capabilities, and calculates the potential security risks. The paper suggests possible risk treatment methods for effective secure system operation. In conclusion, the paper emphasizes the significant benefits of implementing smart irrigation systems, including improved water conservation, increased crop yield, and reduced environmental impact. Additionally, based on the security analysis conducted, the paper recommends the implementation of countermeasures and security approaches to address vulnerabilities and ensure the integrity and reliability of the system. By incorporating these measures, smart irrigation technology can revolutionize water management practices in agriculture, promoting sustainability, resource efficiency, and safeguarding against potential security threats.
Advanced control scheme of doubly fed induction generator for wind turbine us...IJECEIAES
This paper describes a speed control device for generating electrical energy on an electricity network based on the doubly fed induction generator (DFIG) used for wind power conversion systems. At first, a double-fed induction generator model was constructed. A control law is formulated to govern the flow of energy between the stator of a DFIG and the energy network using three types of controllers: proportional integral (PI), sliding mode controller (SMC) and second order sliding mode controller (SOSMC). Their different results in terms of power reference tracking, reaction to unexpected speed fluctuations, sensitivity to perturbations, and resilience against machine parameter alterations are compared. MATLAB/Simulink was used to conduct the simulations for the preceding study. Multiple simulations have shown very satisfying results, and the investigations demonstrate the efficacy and power-enhancing capabilities of the suggested control system.
ACEP Magazine edition 4th launched on 05.06.2024Rahul
This document provides information about the third edition of the magazine "Sthapatya" published by the Association of Civil Engineers (Practicing) Aurangabad. It includes messages from current and past presidents of ACEP, memories and photos from past ACEP events, information on life time achievement awards given by ACEP, and a technical article on concrete maintenance, repairs and strengthening. The document highlights activities of ACEP and provides a technical educational article for members.
Embedded machine learning-based road conditions and driving behavior monitoringIJECEIAES
Car accident rates have increased in recent years, resulting in losses in human lives, properties, and other financial costs. An embedded machine learning-based system is developed to address this critical issue. The system can monitor road conditions, detect driving patterns, and identify aggressive driving behaviors. The system is based on neural networks trained on a comprehensive dataset of driving events, driving styles, and road conditions. The system effectively detects potential risks and helps mitigate the frequency and impact of accidents. The primary goal is to ensure the safety of drivers and vehicles. Collecting data involved gathering information on three key road events: normal street and normal drive, speed bumps, circular yellow speed bumps, and three aggressive driving actions: sudden start, sudden stop, and sudden entry. The gathered data is processed and analyzed using a machine learning system designed for limited power and memory devices. The developed system resulted in 91.9% accuracy, 93.6% precision, and 92% recall. The achieved inference time on an Arduino Nano 33 BLE Sense with a 32-bit CPU running at 64 MHz is 34 ms and requires 2.6 kB peak RAM and 139.9 kB program flash memory, making it suitable for resource-constrained embedded systems.
Optimizing Gradle Builds - Gradle DPE Tour Berlin 2024Sinan KOZAK
Sinan from the Delivery Hero mobile infrastructure engineering team shares a deep dive into performance acceleration with Gradle build cache optimizations. Sinan shares their journey into solving complex build-cache problems that affect Gradle builds. By understanding the challenges and solutions found in our journey, we aim to demonstrate the possibilities for faster builds. The case study reveals how overlapping outputs and cache misconfigurations led to significant increases in build times, especially as the project scaled up with numerous modules using Paparazzi tests. The journey from diagnosing to defeating cache issues offers invaluable lessons on maintaining cache integrity without sacrificing functionality.
Presentation of IEEE Slovenia CIS (Computational Intelligence Society) Chapte...University of Maribor
Slides from talk presenting:
Aleš Zamuda: Presentation of IEEE Slovenia CIS (Computational Intelligence Society) Chapter and Networking.
Presentation at IcETRAN 2024 session:
"Inter-Society Networking Panel GRSS/MTT-S/CIS
Panel Session: Promoting Connection and Cooperation"
IEEE Slovenia GRSS
IEEE Serbia and Montenegro MTT-S
IEEE Slovenia CIS
11TH INTERNATIONAL CONFERENCE ON ELECTRICAL, ELECTRONIC AND COMPUTING ENGINEERING
3-6 June 2024, Niš, Serbia
Introduction- e - waste – definition - sources of e-waste– hazardous substances in e-waste - effects of e-waste on environment and human health- need for e-waste management– e-waste handling rules - waste minimization techniques for managing e-waste – recycling of e-waste - disposal treatment methods of e- waste – mechanism of extraction of precious metal from leaching solution-global Scenario of E-waste – E-waste in India- case studies.
3. TABLE OF CONTENTS
SEMI INFINITE WALL ANALYTICAL SOLUTION TO THE HEAT EQUATION................3
ALTERNATIVE SOLUTION TO THE SEMI-INFINITE WALL PROBLEM ...........................5
HOW DO WE EXPLAIN THE EXISTENCE OF THE FOURIER LAW IN STEADY
STATE FOR SEMI-INFINITE ROD?................................................................................................8
HOW DO WE DEAL WITH CONVECTION AT THE SURFACE AREA OF THE SEMI-
INFINITE METAL ROD ......................................................................................................................11
EQUAL FIXED TEMPERATURES AT THE END OF AN INSULATED METAL ROD....15
UNEQUAL FIXED TEMPERATURES AT THE END OF AN INSULATED METAL ROD.
....................................................................................................................................................................18
HOW DO WE DEAL WITH OTHER TYPES OF BOUNDARY CONDITIONS?..................21
HOW DO WE DEAL WITH NATURAL CONVECTION AT THE SURFACE AREA OF A
SEMI-INFINITE METAL ROD FOR FIXED END TEMPERATURE.....................................23
WHAT HAPPENS WHEN THE INITIAL TEMPERATURE IS A FUNCTION OF X? .......39
HOW DO WE DEAL WITH CYLINDRICAL CO-ORDINATES FOR AN INFINITE
RADIUS CYLINDER? ..........................................................................................................................41
4. SEMI INFINITE WALL ANALYTICAL SOLUTION TO THE
HEAT EQUATION.
The differential equation to be solved is
𝜕𝑇
𝜕𝑡
= 𝛼
𝜕2
𝑇
𝜕𝑥2
Where the initial and boundary conditions are
𝑻 = 𝑻𝒔 𝒂𝒕 𝒙 = 𝟎 𝒇𝒐𝒓 𝒂𝒍𝒍 𝒕
𝑻 = 𝑻∞ 𝒂𝒕 𝒙 = ∞ 𝒇𝒐𝒓 𝒂𝒍𝒍 𝒕
𝑻 = 𝑻∞ 𝒂𝒕 𝒕 = 𝟎 𝒇𝒐𝒓 𝒂𝒍𝒍 𝒙
We postulate:
𝑌 =
𝑇 − 𝑇∞
𝑇𝑠 − 𝑇∞
And
𝜂 =
𝑥
2√𝛼𝑡
We get
𝑑2
𝑌
𝑑𝜂2
+ 2𝜂
𝑑𝑌
𝑑𝜂
= 0 (1)
With the transformed boundary and initial conditions
𝑌 → 0 𝑎𝑠 𝜂 → ∞
And
𝑌 = 1 𝑎𝑡 𝜂 = 0
The first condition is the same as the initial condition 𝑇 = 𝑇∞ 𝑎𝑡 𝑡 = 0 and the
boundary condition
𝑇 → 𝑇∞ 𝑎𝑠 𝑥 → ∞
Equation 1 may be integrated once to get
𝑙𝑛
𝑑𝑌
𝑑𝜂
= 𝑐1 − 𝜂2
𝑑𝑌
𝑑𝜂
= 𝑐2𝑒−𝜂2
5. And integrated once more to get
𝑌 = 𝑐3 + 𝑐2 ∫ 𝑒−𝜂2
𝑑 𝜂
Applying the boundary conditions to the equation, we get
𝑌 = 1 − erf (
𝑥
2√𝛼𝑡
)
𝑻 − 𝑻∞
𝑻𝒔 − 𝑻∞
= 𝟏 − 𝐞𝐫𝐟 (
𝒙
𝟐√𝜶𝒕
)
Or
𝑻𝒔 − 𝑻
𝑻𝒔 − 𝑻∞
= 𝐞𝐫𝐟 (
𝒙
𝟐√𝜶𝒕
)
6. ALTERNATIVE SOLUTION TO THE SEMI-INFINITE WALL PROBLEM
The problem of the semi-infinite wall could also be solved as below:
Given the boundary and initial conditions
𝑻 = 𝑻𝒔 𝒂𝒕 𝒙 = 𝟎 𝒇𝒐𝒓 𝒂𝒍𝒍 𝒕
𝑻 = 𝑻∞ 𝒂𝒕 𝒙 = ∞
𝑻 = 𝑻∞ 𝒂𝒕 𝒕 = 𝟎
Since the rod is infinite in one direction, we say 𝑙 = ∞
And the governing equation
𝜕𝑇
𝜕𝑡
= 𝛼
𝜕2
𝑇
𝜕𝑥2
We assume an exponential temperature profile that satisfies the boundary
conditions:
𝑇 − 𝑇∞
𝑇𝑠 − 𝑇∞
= 𝑒
−𝑥
𝛿
We can satisfy the initial condition if we assume that 𝛿 will have a solution as
𝛿 = 𝑐𝑡𝑛
Where c and n are constants so that at 𝑡 = 0, 𝛿 = 0 and the initial condition is
satisfied as shown below.
𝑇 − 𝑇∞
𝑇𝑠 − 𝑇∞
= 𝑒
−𝑥
0 = 𝑒−∞
= 0
𝑇 = 𝑇∞ 𝑎𝑡 𝑡 = 0
We then transform the heat governing equation into an integral equation as:
𝛼 ∫ (
𝜕2
𝑇
𝜕𝑥2
) 𝑑𝑥
𝑙
0
= ∫
𝜕𝑇
𝜕𝑡
𝑑𝑥
𝑙
0
𝛼 ∫ (
𝜕2
𝑇
𝜕𝑥2
) 𝑑𝑥
𝑙
0
=
𝜕
𝜕𝑡
∫ 𝑇𝑑𝑥
𝑙
0
𝑇 = (𝑇𝑠 − 𝑇∞)𝑒
−𝑥
𝛿 + 𝑇∞
𝜕
𝜕𝑡
∫ 𝑇𝑑𝑥
𝑙
0
=
𝜕
𝜕𝑡
∫ ((𝑇𝑠 − 𝑇∞)𝑒
−𝑥
𝛿 + 𝑇∞)𝑑𝑥 =
𝜕
𝜕𝑡
[𝛿(𝑇𝑠 − 𝑇∞)(1 − 𝑒
−𝑙
𝛿 )] −
𝜕(𝑙𝑇∞)
𝜕𝑡
𝑙
0
Since 𝑙 and 𝑇∞ are constants independent of time
7. 𝜕(𝑙𝑇∞)
𝜕𝑡
= 0
So
𝜕
𝜕𝑡
∫ 𝑇𝑑𝑥
𝑙
0
=
𝜕
𝜕𝑡
[𝛿(𝑇𝑠 − 𝑇∞)(1 − 𝑒
−𝑙
𝛿 )]
Since 𝑙 = ∞, we substitute for 𝑙 and get
𝜕
𝜕𝑡
∫ 𝑇𝑑𝑥
𝑙
0
=
𝑑𝛿
𝑑𝑡
(𝑇𝑠 − 𝑇∞)
We go ahead and find
𝜕2
𝑇
𝜕𝑥2
=
(𝑇𝑠 − 𝑇∞)
𝛿2
∫ (
𝜕2
𝑇
𝜕𝑥2
) 𝑑𝑥
𝑙
0
= −
(𝑇𝑠 − 𝑇∞)
𝛿
(𝑒
−𝑙
𝛿 − 1) =
(𝑇𝑠 − 𝑇∞)
𝛿
(1 − 𝑒
−𝑙
𝛿 )
Since 𝑙 = ∞, we substitute for 𝑙 and get
∫ (
𝜕2
𝑇
𝜕𝑥2
) 𝑑𝑥
𝑙
0
= −
(𝑇𝑠 − 𝑇∞)
𝛿
(𝑒
−𝑙
𝛿 − 1) =
(𝑇𝑠 − 𝑇∞)
𝛿
Substituting into the integral equation, we get
𝛼
(𝑇𝑠 − 𝑇∞)
𝛿
=
𝑑𝛿
𝑑𝑡
(𝑇𝑠 − 𝑇∞)
The boundary conditions are
𝛿 = 0 𝑎𝑡 𝑡 = 0
We find
𝛿 = √2𝛼𝑡
We substitute in the temperature profile and get
𝑻 − 𝑻∞
𝑻𝒔 − 𝑻∞
= 𝒆
−𝒙
√𝟐𝜶𝒕
You notice that the initial condition is satisfied by the temperature profile
above i.e.,
8. At 𝑡 = 0
𝑻 − 𝑻∞
𝑻𝒔 − 𝑻∞
= 𝒆
−𝒙
√𝟐𝜶𝒕
Becomes
𝑻 − 𝑻∞
𝑻𝒔 − 𝑻∞
= 𝒆
−𝒙
𝟎 = 𝒆−∞
= 𝟎
Hence 𝑻 = 𝑻∞ throughout the rod at 𝑡 = 0
Observation.
The two equations
𝑻 − 𝑻∞
𝑻𝒔 − 𝑻∞
= 𝒆
−𝒙
√𝟐𝜶𝒕
And
𝑻 − 𝑻∞
𝑻𝒔 − 𝑻∞
= 𝟏 − 𝐞𝐫𝐟 (
𝒙
𝟐√𝜶𝒕
)
Should give the same answer. Indeed, they give answers that are the same with
a small error since the error function is got from tables after rounding off yet in
the exponential temperature profile there is no rounding off.
9. HOW DO WE EXPLAIN THE EXISTENCE OF THE
FOURIER LAW IN STEADY STATE FOR SEMI-INFINITE
ROD?
The Fourier law states:
𝑄 = −𝑘𝐴
𝜕𝑇
𝜕𝑥
Under steady state.
It can be stated as:
𝜕𝑇
𝜕𝑥
= −
𝑄
𝑘𝐴
Under steady state.
To satisfy the Fourier law under steady state, we postulate the temperature
profile to be:
𝑻 − 𝑻∞ =
𝑸
𝒌𝑨
𝜹𝒆
−𝒙
𝜹
𝛿 is a function of time 𝑡 and not distance 𝑥
We believe that after solving for 𝛿, 𝛿 will be directly proportional to time t so
that 𝛿 = 𝑘𝑡𝑛
sothat at 𝑡 = ∞ , 𝛿 = ∞
And taking the first derivative of temperature with distance x at 𝑡 = ∞ , we get
𝜕𝑇
𝜕𝑥
|𝑡=∞ = −
𝑄
𝑘𝐴
𝑒
−𝑥
𝛿 = −
𝑄
𝑘𝐴
𝑒
−𝑥
∞ = −
𝑄
𝑘𝐴
𝑒0
𝝏𝑻
𝝏𝒙
= −
𝑸
𝒌𝑨
Hence the Fourier law is satisfied.
Now let us go ahead and solve for 𝛿.
Recall
PDE
𝜕𝑇
𝜕𝑡
= 𝛼
𝜕2
𝑇
𝜕𝑥2
The initial condition is
𝑇 = 𝑇∞ 𝑎𝑡 𝑡 = 0
10. The boundary conditions are
𝑇 = 𝑇∞ 𝑎𝑡 𝑥 = ∞
𝜕𝑇
𝜕𝑥
|𝑥=0 = −
𝑄
𝑘𝐴
The temperature profile that satisfies the conditions above is
𝑇 − 𝑇∞ =
𝑄
𝑘𝐴
𝛿𝑒
−𝑥
𝛿
We transform the PDE into an integral equation by integrating over the whole
length of the metal rod.
𝜕
𝜕𝑡
∫ (𝑇)𝑑𝑥
𝑙
0
= 𝛼 ∫ (
𝜕2
𝑇
𝜕𝑥2
) 𝑑𝑥
𝑙
0
And using the temperature profile, we get
∫ (
𝜕2
𝑇
𝜕𝑥2
) 𝑑𝑥
𝑙
0
= [
𝜕𝑇
𝜕𝑥
]
𝑙
0
=
𝑄
𝑘𝐴
(1 − 𝑒
−𝑙
𝛿 )
Since 𝑙 = ∞, we substitute for 𝑙 and get
∫ (
𝜕2
𝑇
𝜕𝑥2
) 𝑑𝑥
𝑙
0
= [
𝜕𝑇
𝜕𝑥
]
𝑙
0
=
𝑄
𝑘𝐴
𝑇 =
𝑄
𝑘𝐴
𝛿𝑒
−𝑥
𝛿 + 𝑇∞
𝜕
𝜕𝑡
∫ (𝑇)𝑑𝑥
𝑙
0
=
𝜕
𝜕𝑡
∫ (
𝑄
𝑘𝐴
𝛿𝑒
−𝑥
𝛿 + 𝑇∞)𝑑𝑥
𝑙
0
=
𝜕
𝜕𝑡
[
𝑄
𝑘𝐴
𝛿2
(1 − 𝑒
−𝑙
𝛿 )] +
𝜕(𝑙𝑇∞)
𝜕𝑡
𝜕(𝑙𝑇∞)
𝜕𝑡
= 0
𝜕
𝜕𝑡
∫ (𝑇)𝑑𝑥
𝑙
0
=
𝜕
𝜕𝑡
[
𝑄
𝑘𝐴
𝛿2
(1 − 𝑒
−𝑙
𝛿 )]
Since 𝑙 = ∞, we substitute for 𝑙 and get
𝜕
𝜕𝑡
∫ (𝑇)𝑑𝑥
𝑙
0
=
𝜕
𝜕𝑡
[
𝑄
𝑘𝐴
𝛿2
]
We then substitute into the integral equation
11. 𝛼
𝑄
𝑘𝐴
= 2𝛿
𝑑𝛿
𝑑𝑡
(
𝑄
𝑘𝐴
)
The boundary conditions are
𝛿 = 0 𝑎𝑡 𝑡 = 0
𝛿 = √𝛼𝑡
Substituting into the temperature profile, we get
𝑇 − 𝑇∞ =
𝑄
𝑘𝐴
𝛿𝑒
−𝑥
𝛿
𝑻 − 𝑻∞ =
𝑸
𝒌𝑨
× √𝜶𝒕 × 𝒆
−𝒙
√𝜶𝒕
You notice that the initial condition is satisfied
𝝏𝑻
𝝏𝒙
|𝒕=∞ = −
𝑸
𝒌𝑨
Hence the Fourier law
So, our assumption of 𝛿 = 𝑘𝑡𝑛
is satisfied
12. HOW DO WE DEAL WITH CONVECTION AT THE
SURFACE AREA OF THE SEMI-INFINITE METAL ROD
Recall that the temperature profile that satisfies the Fourier law was
𝑇 − 𝑇∞ =
𝑄
𝑘𝐴
𝛿𝑒
−𝑥
𝛿
Recall
PDE
𝛼
𝜕2
𝑇
𝜕𝑥2
−
ℎ𝑃
𝐴𝜌𝐶
(𝑇 − 𝑇∞) =
𝜕𝑇
𝜕𝑡
The initial condition is
𝑇 = 𝑇∞ 𝑎𝑡 𝑡 = 0
The boundary conditions are
𝑇 = 𝑇∞ 𝑎𝑡 𝑥 = ∞
𝜕𝑇
𝜕𝑥
|𝑥=0 = −
𝑄
𝑘𝐴
Remember that for a semi-infinite rod 𝑙 = ∞
We transform the PDE into an integral equation
𝛼 ∫ (
𝜕2
𝑇
𝜕𝑥2
) 𝑑𝑥
𝑙
0
−
ℎ𝑃
𝐴𝜌𝐶
∫ (𝑇 − 𝑇∞)𝑑𝑥
𝑙
0
=
𝜕
𝜕𝑡
∫ (𝑇)𝑑𝑥
𝑙
0
Where:
We are dealing with a cylindrical metal rod.
𝑃 = 2𝜋𝑟 𝑎𝑛𝑑 𝜌 = 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑚𝑒𝑡𝑎𝑙 𝑟𝑜𝑑
And using the temperature profile, we get
∫ (
𝜕2
𝑇
𝜕𝑥2
) 𝑑𝑥
𝑙
0
=
𝑄
𝑘𝐴
(1 − 𝑒
−𝑙
𝛿 )
Substitute for 𝑙 = ∞ and get
∫ (
𝜕2
𝑇
𝜕𝑥2
) 𝑑𝑥
𝑙
0
=
𝑄
𝑘𝐴
13. ∫ (𝑇)𝑑𝑥
𝑙
0
=
𝑄
𝑘𝐴
𝛿2
(1 − 𝑒
−𝑙
𝛿 ) + 𝑇∞𝑙
𝜕
𝜕𝑡
∫ (𝑇)𝑑𝑥
𝑙
0
=
𝜕
𝜕𝑡
(
𝑄
𝑘𝐴
𝛿2
(1 − 𝑒
−𝑙
𝛿 )) +
𝜕(𝑇∞𝑙)
𝜕𝑡
𝜕(𝑇∞𝑙)
𝜕𝑡
= 0
𝜕
𝜕𝑡
∫ (𝑇)𝑑𝑥
𝑙
0
=
𝜕
𝜕𝑡
(
𝑄
𝑘𝐴
𝛿2
(1 − 𝑒
−𝑙
𝛿 ))
Substitute for 𝑙 = ∞ and get
𝜕
𝜕𝑡
∫ (𝑇)𝑑𝑥
𝑙
0
=
𝜕
𝜕𝑡
(
𝑄
𝑘𝐴
𝛿2
) = 2𝛿(
𝑄
𝑘𝐴
)
𝑑𝛿
𝑑𝑡
∫ (𝑇 − 𝑇∞)𝑑𝑥
𝑙
0
=
𝑄
𝑘𝐴
𝛿2
(1 − 𝑒
−𝑙
𝛿 )
Substitute for 𝑙 = ∞ and get
∫ (𝑇 − 𝑇∞)𝑑𝑥
𝑙
0
=
𝑄
𝑘𝐴
𝛿2
Substituting in the integral equation above, we get
𝛼 −
ℎ𝑃
𝐴𝜌𝐶
𝛿2
= 2𝛿
𝑑𝛿
𝑑𝑡
The boundary condition is
𝛿 = 0 𝑎𝑡 𝑡 = 0
We solve and get
𝛿 = √
𝐴𝜌𝐶𝛼
ℎ𝑃
(1 − 𝑒
−ℎ𝑃𝑡
𝐴𝜌𝐶 )
Substituting in the temperature profile, we get
𝑇 − 𝑇∞ =
𝑄
𝑘𝐴
𝛿𝑒
−𝑥
𝛿
𝑻 − 𝑻∞ =
𝑸
𝒌𝑨
× √
𝑨𝝆𝑪𝜶
𝒉𝑷
(𝟏 − 𝒆
−𝒉𝑷𝒕
𝑨𝝆𝑪 ) × 𝒆
−𝒙
√𝑨𝝆𝑪
𝒉𝑷
(𝟏−𝒆
−𝒉𝑷𝒕
𝑨𝝆𝑪 )
14. We notice that the initial condition and boundary conditions are satisfied.
For small time the term
ℎ𝑃𝑡
𝐴𝜌𝐶
≪ 1
And using binomial approximation of the exponential, we get
𝑒
−ℎ𝑃𝑡
𝐴𝜌𝐶 = 1 −
ℎ𝑃𝑡
𝐴𝜌𝐶
Then
(1 − 𝑒
−ℎ𝑃𝑡
𝐴𝜌𝐶 ) =
ℎ𝑃𝑡
𝐴𝜌𝐶
Upon substitution in the temperature profile, we get
𝑇 − 𝑇∞ =
𝑄
𝑘𝐴
× √𝛼𝑡 × 𝑒
−𝑥
√𝛼𝑡
Upon rearranging, we get
𝑥
√𝛼𝑡
= ln (
𝑄
𝑘𝐴
√𝛼𝑡) − ln (𝑇 − 𝑇∞)
𝑥
√𝑡
= √𝛼ln(√𝑡) + √𝛼 [ln (
𝑄
𝑘𝐴
√𝛼) − ln(𝑇 − 𝑇∞)]
What we observe is
𝒙
√𝒕
= √𝜶𝐥𝐧(√𝒕) + √𝜶 [𝐥𝐧 (
𝑸
𝒌𝑨√𝜶
(𝑻 − 𝑻∞)
)]
That is what we observe for short times.
When the times become big, we observe
𝑻 − 𝑻∞ =
𝑸
𝒌𝑨
× √
𝑨𝝆𝑪𝜶
𝒉𝑷
(𝟏 − 𝒆
−𝒉𝑷𝒕
𝑨𝝆𝑪 ) × 𝒆
−𝒙
√𝑨𝝆𝑪
𝒉𝑷
(𝟏−𝒆
−𝒉𝑷𝒕
𝑨𝝆𝑪 )
15. And in steady state (𝑡 = ∞), we observe
𝑻 − 𝑻∞ =
𝑸
𝒌𝑨
× √
𝑨𝝆𝑪𝜶
𝒉𝑷
× 𝒆
−𝒙
√𝑨𝝆𝑪𝜶
𝒉𝑷
𝛼 =
𝑘
𝜌𝐶
We finally get
𝑻 − 𝑻∞ =
𝑸
𝒌𝑨
× √
𝒌𝑨
𝒉𝑷
× 𝒆
−𝒙
√𝒌𝑨
𝒉𝑷
the heat flow in steady state is given by:
𝜕𝑇
𝜕𝑥
= −
𝑄
𝑘𝐴
𝑒
−𝑥
√𝑘𝐴
ℎ𝑃
−𝒌𝑨
𝝏𝑻
𝝏𝒙
= 𝑸𝒆
−𝒙
√𝒌𝑨
𝒉𝑷
16. EQUAL FIXED TEMPERATURES AT THE END OF AN
INSULATED METAL ROD.
PDE
𝜕𝑇
𝜕𝑡
= 𝛼
𝜕2
𝑇
𝜕𝑥2
BCs
𝑻 = 𝑻𝒔 𝒂𝒕 𝒙 = 𝟎 𝟎 < 𝒕 < ∞
𝑻 = 𝑻𝒔 𝒂𝒕 𝒙 = 𝒍 𝟎 < 𝒕 < ∞
IC
𝑻 = 𝑻∞ 𝒂𝒕 𝒕 = 𝟎 𝟎 ≤ 𝒙 ≤ 𝒍
we know a Fourier series solution exists given by
𝑻 − 𝑻𝒔
𝑻∞ − 𝑻𝒔
=
𝟒
𝝅
∑
𝟏
𝒏
∞
𝒏=𝟏
𝒔𝒊𝒏 (
𝒏𝝅𝒙
𝒍
) 𝒆
−(
𝒏𝝅
𝟐 )
𝜶𝒕
(
𝒍
𝟐
)𝟐
𝒏 = 𝟏, 𝟑, 𝟓, …
You notice that this solution is not entirely deterministic since it involves
summing terms up to infinity.
There is an alternative solution as shown below:
𝜕𝑇
𝜕𝑡
= 𝛼
𝜕2
𝑇
𝜕𝑥2
BCs
𝑻 = 𝑻𝒔 𝒂𝒕 𝒙 = 𝟎 𝟎 < 𝒕 < ∞
𝑻 = 𝑻𝒔 𝒂𝒕 𝒙 = 𝒍 𝟎 < 𝒕 < ∞
IC
𝑻 = 𝑻∞ 𝒂𝒕 𝒕 = 𝟎 𝟎 ≤ 𝒙 ≤ 𝒍
We assume an exponential temperature profile that satisfies the boundary
conditions:
𝑇 − 𝑇∞
𝑇𝑠 − 𝑇∞
= 𝑒
−𝑥
𝛿
(1−
𝑥
𝑙
)
17. You notice that the temperature profile above satisfies the boundary
conditions. We can satisfy the initial condition if we assume that 𝛿 will assume
a solution as
𝛿 = 𝑐𝑡𝑛
Where c and n are constants so that at 𝑡 = 0, 𝛿 = 0 and the initial condition is
satisfied as shown below.
𝑇 − 𝑇∞
𝑇𝑠 − 𝑇∞
= 𝑒
−𝑥
0 = 𝑒−∞
= 0
𝑇 = 𝑇∞ 𝑎𝑡 𝑡 = 0
We transform the PDE into an integral equation
𝜕
𝜕𝑡
∫ (𝑇)𝑑𝑥
𝑙
0
= 𝛼 ∫ (
𝜕2
𝑇
𝜕𝑥2
) 𝑑𝑥
𝑙
0
∫ (
𝜕2
𝑇
𝜕𝑥2
) 𝑑𝑥
𝑙
0
= [
𝜕𝑇
𝜕𝑥
]
𝑙
0
= (𝑇𝑠 − 𝑇∞) [
(−𝑙 + 2𝑥)
𝛿𝑙
× 𝑒
−𝑥
𝛿
(1−
𝑥
𝑙
)
] 𝑙
0
∫ (
𝜕2
𝑇
𝜕𝑥2
) 𝑑𝑥
𝑙
0
= (𝑇𝑠 − 𝑇∞) [
(−𝑙 + 2𝑥)
𝛿𝑙
× 𝑒
−𝑥
𝛿
(1−
𝑥
𝑙
)
] 𝑙
0
=
2(𝑇𝑠 − 𝑇∞)
𝛿
𝑇 = (𝑇𝑠 − 𝑇∞)𝑒
−𝑥
𝛿
(1−
𝑥
𝑙
)
+ 𝑇∞
∫ (𝑇)𝑑𝑥
𝑙
0
= (𝑇𝑠 − 𝑇∞) [
𝛿𝑙
(−𝑙 + 2𝑥)
× 𝑒
−𝑥
𝛿
(1−
𝑥
𝑙
)
] 𝑙
0
+ 𝑇∞𝑙 = 2(𝑇𝑠 − 𝑇∞)𝛿 + 𝑇∞𝑙
𝜕
𝜕𝑡
∫ (𝑇)𝑑𝑥
𝑙
0
= 2(𝑇𝑠 − 𝑇∞)
𝑑𝛿
𝑑𝑡
+
𝑑(𝑇∞𝑙)
𝑑𝑡
𝑑(𝑇∞𝑙)
𝑑𝑡
= 0
𝜕
𝜕𝑡
∫ (𝑇)𝑑𝑥
𝑙
0
= 2(𝑇𝑠 − 𝑇∞)
𝑑𝛿
𝑑𝑡
Substituting in the integral equation above, we get:
𝛼 (
2
𝛿
) = 2
𝑑𝛿
𝑑𝑡
𝛿 = √2𝛼𝑡
Substituting back 𝛿 into the temperature profile, we get
18. 𝑻 − 𝑻∞
𝑻𝒔 − 𝑻∞
= 𝒆
−𝒙
√𝟐𝜶𝒕
(𝟏−
𝒙
𝒍
)
Or
𝑻 − 𝑻𝒔
𝑻∞ − 𝑻𝒔
= 𝟏 − 𝒆
−𝒙
√𝟐𝜶𝒕
(𝟏−
𝒙
𝒍
)
You notice that the initial condition is satisfied.
You notice that when 𝑙 = ∞ , we reduce to the temperature profile we derived
before
𝑻 − 𝑻∞
𝑻𝒔 − 𝑻∞
= 𝒆
−𝒙
√𝟐𝜶𝒕
you notice that in the temperature profile developed, we get an exact solution
to the problem not an approximate as the Fourier series.
19. UNEQUAL FIXED TEMPERATURES AT THE END OF AN
INSULATED METAL ROD.
𝜕𝑇
𝜕𝑡
= 𝛼
𝜕2
𝑇
𝜕𝑥2
The boundary conditions are:
𝑇 = 𝑇𝑠 𝑎𝑡 𝑥 = 0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡
𝑇 = 𝑇1 𝑎𝑡 𝑥 = 𝑙 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡
The initial condition is
𝑇 = 𝑇∞ 𝑎𝑡 𝑡 = 0 0 ≤ 𝑥 ≤ 𝑙
The temperature profile that satisfies the boundary conditions is:
𝑻 − 𝑻∞
[
𝒙
𝒍
(𝑻𝟏 − 𝑻∞) + (𝑻𝒔 − 𝑻∞) (𝟏 −
𝒙
𝒍)]
= 𝒆−
𝒙
𝜹
(𝟏−
𝒙
𝒍
)
For now, we shall have a solution where 𝛿 is proportional to time t so that at
𝑡 = 0, 𝛿 = 0 and the initial condition will be satisfied.
We then transform the heat governing equation into an integral equation as:
𝛼 ∫ (
𝜕2
𝑇
𝜕𝑥2
) 𝑑𝑥
𝑙
0
= ∫
𝜕𝑇
𝜕𝑡
𝑑𝑥
𝑙
0
Where:
𝑙 = 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑡ℎ𝑒 𝑚𝑒𝑡𝑎𝑙 𝑟𝑜𝑑
So, the integral equation becomes:
𝛼 ∫ (
𝜕2
𝑇
𝜕𝑥2
) 𝑑𝑥
𝑙
0
=
𝜕
𝜕𝑡
∫ (𝑇)𝑑𝑥
𝑙
0
We go ahead and find
∫ (
𝜕2
𝑇
𝜕𝑥2
) 𝑑𝑥
𝑙
0
= [
𝜕𝑇
𝜕𝑥
]
𝑙
0
𝜕𝑇
𝜕𝑥
= (𝑇𝑠 − 𝑇∞)𝑒−
𝑥
𝛿
(1−
𝑥
𝑙
)
[
1
𝑙
(𝑇1 − 𝑇∞)
(𝑇𝑠 − 𝑇∞)
−
1
𝑙
] + (𝑇𝑠 − 𝑇∞)𝑒−
𝑥
𝛿
(1−
𝑥
𝑙
)
[
𝑥
𝑙
(𝑇1 − 𝑇∞)
(𝑇𝑠 − 𝑇∞)
− (1 −
𝑥
𝑙
)] (
−𝑙 + 2𝑥
𝛿𝑙
)
22. HOW DO WE DEAL WITH OTHER TYPES OF BOUNDARY
CONDITIONS?
Consider the following types of boundary conditions and initial condition:
A)
𝑻 = 𝑻𝒔 𝒂𝒕 𝒙 = 𝟎
𝒅𝑻
𝒅𝒙
= 𝟎 𝒂𝒕 𝒙 = 𝒍
𝑻 = 𝑻∞ 𝒂𝒕 𝒕 = 𝟎
B)
𝑻 = 𝑻𝒔 𝒂𝒕 𝒙 = 𝟎
−𝒌
𝒅𝑻
𝒅𝒙
= 𝒉(𝑻 − 𝑻∞) 𝒂𝒕 𝒙 = 𝒍
𝑻 = 𝑻∞ 𝒂𝒕 𝒕 = 𝟎
Let us go about solving for the above boundary conditions but let us deal with set A
boundary conditions and then we can deal with set B later.
We start with a temperature profile below:
𝑇 − 𝑇∞ = (𝑇𝑠 − 𝑇∞)𝑒−
𝑥
𝛿
(1−
𝑥
𝑙
)
[
𝑥
𝑙
(𝑇1 − 𝑇∞)
(𝑇𝑠 − 𝑇∞)
+ (1 −
𝑥
𝑙
)]
we take the derivative
𝒅𝑻
𝒅𝒙
𝒂𝒕 𝒙 = 𝒍 and equate it to 0 and get:
𝑑𝑇
𝑑𝑥
|𝑥=𝑙 = (
(𝑇1 − 𝑇∞)
𝑙
−
(𝑇𝑠 − 𝑇∞)
𝑙
+
(𝑇1 − 𝑇∞)
𝛿
)
𝑑𝑇
𝑑𝑥
|𝑥=𝑙 = (
(𝑇1 − 𝑇∞)
𝑙
−
(𝑇𝑠 − 𝑇∞)
𝑙
+
(𝑇1 − 𝑇∞)
𝛿
) = 0
We finally get
(𝑇1 − 𝑇∞) = (𝑇𝑠 − 𝑇∞)(
𝛿
𝑙 + 𝛿
)
We substitute 𝑇1 − 𝑇∞ into the temperature profile and get
(𝑻 − 𝑻∞) = (𝑻𝒔 − 𝑻∞)𝒆−
𝒙
𝜹
(𝟏 −
𝒙
𝒍
)
[
𝒙
𝒍
(
𝜹
𝜹 + 𝒍
) + (𝟏 −
𝒙
𝒍
)]
23. So, the temperature profile above satisfies the set A) boundary and initial
conditions and we can go ahead and solve the governing equation using the
temperature profile above.
For set B) boundary conditions, we again start with the temperature profile
below:
𝑇 − 𝑇∞ = (𝑇𝑠 − 𝑇∞)𝑒−
𝑥
𝛿
(1 −
𝑥
𝑙
)
[
𝑥
𝑙
(𝑇1 − 𝑇∞)
(𝑇𝑠 − 𝑇∞)
+ (1 −
𝑥
𝑙
)]
we take the derivative
𝒅𝑻
𝒅𝒙
𝒂𝒕 𝒙 = 𝒍 and equate it to:
𝑑𝑇
𝑑𝑥
|𝑥=𝑙 = (
(𝑇1 − 𝑇∞)
𝑙
−
(𝑇𝑠 − 𝑇∞)
𝑙
+
(𝑇1 − 𝑇∞)
𝛿
)
𝒅𝑻
𝒅𝒙
|𝒙=𝒍 = −
𝒉
𝒌
(𝑻𝟏 − 𝑻∞)
We then find the required temperature profile which we can use to solve the
governing equation.
24. HOW DO WE DEAL WITH NATURAL CONVECTION AT
THE SURFACE AREA OF A SEMI-INFINITE METAL ROD
FOR FIXED END TEMPERATURE
𝛼
𝜕2
𝑇
𝜕𝑥2
−
ℎ𝑃
𝐴𝜌𝐶
(𝑇 − 𝑇∞) =
𝜕𝑇
𝜕𝑡
We shall use the integral approach.
The boundary and initial conditions are
𝑻 = 𝑻𝒔 𝒂𝒕 𝒙 = 𝟎 𝒇𝒐𝒓 𝒂𝒍𝒍 𝒕
𝑻 = 𝑻∞ 𝒂𝒕 𝒙 = ∞
𝑻 = 𝑻∞ 𝒂𝒕 𝒕 = 𝟎
Where: 𝑻∞ = 𝒓𝒐𝒐𝒎 𝒕𝒆𝒎𝒑𝒆𝒓𝒂𝒕𝒖𝒓𝒆
First, we assume a temperature profile that satisfies the boundary conditions as:
𝑇 − 𝑇∞
𝑇𝑠 − 𝑇∞
= 𝑒
−𝑥
𝛿
where 𝛿 is to be determined and is a function of time t.
The governing equation is
𝛼
𝜕2
𝑇
𝜕𝑥2
−
ℎ𝑃
𝐴𝜌𝐶
(𝑇 − 𝑇∞) =
𝜕𝑇
𝜕𝑡
Let us change this equation into an integral as below:
𝛼 ∫ (
𝜕2
𝑇
𝜕𝑥2
) 𝑑𝑥
𝑙
0
−
ℎ𝑃
𝐴𝜌𝐶
∫ (𝑇 − 𝑇∞)𝑑𝑥
𝑙
0
=
𝜕
𝜕𝑡
∫ (𝑇)𝑑𝑥
𝑙
0
… … . . 𝑏)
𝜕2
𝑇
𝜕𝑥2
=
(𝑇𝑠 − 𝑇∞)
𝛿2
𝑒
−𝑥
𝛿
∫ (
𝜕2
𝑇
𝜕𝑥2
) 𝑑𝑥
𝑙
0
=
−(𝑇𝑠 − 𝑇∞)
𝛿
(𝑒
−𝑙
𝛿 − 1)
But 𝑙 = ∞, upon substitution, we get
∫ (
𝜕2
𝑇
𝜕𝑥2
) 𝑑𝑥
𝑙
0
=
(𝑇𝑠 − 𝑇∞)
𝛿
25. ∫ (𝑇 − 𝑇∞)𝑑𝑥
𝑙
0
= −𝛿(𝑇𝑠 − 𝑇∞)(𝑒
−𝑙
𝛿 − 1)
But 𝑙 = ∞, upon substitution, we get
∫ (𝑇 − 𝑇∞)𝑑𝑥
𝑙
0
= 𝛿(𝑇𝑠 − 𝑇∞)
∫ (𝑇)𝑑𝑥
𝑙
0
= 𝛿(𝑇𝑠 − 𝑇∞)(𝑒
−𝑙
𝛿 − 1) + 𝑇∞𝑙
Substitute 𝑙 = ∞ and get
𝜕
𝜕𝑡
∫ (𝑇)𝑑𝑥
𝑙
0
=
𝑑𝛿
𝑑𝑡
(𝑇𝑠 − 𝑇∞) +
𝜕
𝜕𝑡
(𝑇∞𝑙)
𝜕
𝜕𝑡
(𝑇∞𝑙) = 0
𝜕
𝜕𝑡
∫ (𝑇)𝑑𝑥
𝑙
0
=
𝑑𝛿
𝑑𝑡
(𝑇𝑠 − 𝑇∞)
Substituting the above expressions in equation b) above, we get
𝛼 −
ℎ𝑃
𝐴𝜌𝐶
𝛿2
= 𝛿
𝑑𝛿
𝑑𝑡
We solve the equation above assuming that
𝛿 = 0 𝑎𝑡 𝑡 = 0
And get
𝛿 = √
𝛼𝐴𝜌𝐶
ℎ𝑃
(1 − 𝑒
−2ℎ𝑃
𝐴𝜌𝐶
𝑡
)
𝛿 = √
𝐾𝐴
ℎ𝑃
(1 − 𝑒
−2ℎ𝑃
𝐴𝜌𝐶
𝑡
)
Substituting for 𝛿 in the temperature profile, we get
26. 𝑻 − 𝑻∞
𝑻𝒔 − 𝑻∞
= 𝒆
−𝒙
√𝑲𝑨
𝒉𝑷
(𝟏−𝒆
−𝟐𝒉𝑷
𝑨𝝆𝑪
𝒕
)
From the equation above, we notice that the initial condition is satisfied i.e.,
𝑻 = 𝑻∞ 𝒂𝒕 𝒕 = 𝟎
The equation above predicts the transient state and in steady state (𝑡 = ∞) it
reduces to
𝑻 − 𝑻∞
𝑻𝒔 − 𝑻∞
= 𝒆
−√(
𝒉𝑷
𝑲𝑨
)𝒙
What are the predictions of the transient state?
Let us make 𝑥 the subject of the equation of transient state and get:
𝑥2
= [ln (
𝑇𝑠 − 𝑇∞
𝑇 − 𝑇∞
)]2
×
𝐾𝐴
ℎ𝑃
(1 − 𝑒
−2ℎ𝑃
𝐴𝜌𝐶
𝑡
)
When the time duration is small and
2ℎ𝑃
𝐴𝜌𝐶
𝑡 ≪ 1
We use the binomial expansion approximation
𝑒
−2ℎ𝑃
𝐴𝜌𝐶
𝑡
= 1 −
2ℎ𝑃
𝐴𝜌𝐶
𝑡
Substituting in the equation of 𝑥2
as the subject, we get
𝑥2
= 2𝛼[ln (
𝑇𝑠 − 𝑇∞
𝑇 − 𝑇∞
)]2
× 𝑡
Where:
𝛼 =
𝐾
𝜌𝐶
We can include an intercept term 𝑡0 which is observed experimentally i.e.,
𝑥2
= 2𝛼[ln (
𝑇𝑠 − 𝑇∞
𝑇 − 𝑇∞
)]2
× (𝑡 − 𝑡0)
Where:
27. 𝑡0 = 𝑎
𝑟𝜌𝐶
2ℎ
And
𝑎 = 𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 = 2.5926 × 10−6
The above implies that
𝛿 = 0 𝑎𝑡 𝑡 = 𝑡0
That there is a lag in the motion of the heat boundary layer by a time 𝑡0.
The equation becomes
𝒙𝟐
= 𝟐𝜶[𝐥𝐧 (
𝑻𝒔 − 𝑻∞
𝑻 − 𝑻∞
)]𝟐
× 𝒕 − 𝟐𝜶[𝐥𝐧 (
𝑻𝒔 − 𝑻∞
𝑻 − 𝑻∞
)]𝟐
× 𝒂
𝒓𝝆𝑪
𝟐𝒉
Where:
𝛼 =
𝐾
𝜌𝐶
The equation becomes:
𝒙𝟐
= 𝟐𝜶[𝐥𝐧 (
𝑻𝒔 − 𝑻∞
𝑻 − 𝑻∞
)]𝟐
× 𝒕 −
𝒂𝑲𝒓
𝒉
[𝐥𝐧 (
𝑻𝒔 − 𝑻∞
𝑻 − 𝑻∞
)]𝟐
What that equation says is that when you stick wax particles on a long metal
rod (𝑙 = ∞) at distances x from the hot end of the rod and note the time t it
takes the wax particles to melt, then a graph of 𝑥2
against 𝑡 is a straight-line
graph with an intercept as stated by the equation above when the times are
small. The equation is true because that is what is observed experimentally.
The intercept above leads to an increase in time of flow of a boundary layer.
28. Since the graph above is a straight-line graph, it shows that 𝑇𝑠 IS NOT a
function of time.
In the equation above we substitute 𝑇 = 37℃ which is the temperature at which
wax begins to melt.
You notice that by varying the radius of the rod and plotting a graph of 𝑥2
against time t for melting wax at the sides of the rod, from the intercept, the
constant ‘a’ above can be measured and from the gradient, 𝑇𝑠 can be measured
since 𝑇 = 37℃ in the equation.
For an aluminium rod of radius 2mm, 𝑇𝑠 was found to be 57℃.
NB
• The temperature at which wax begins to melt is 37℃
• From experiment, it was found that 𝑇𝑠 is not the temperature of the flame
at the beginning of the metal rod.
To get 𝑇𝑠 we plot the graph of
𝑇 − 𝑇∞
𝑇𝑠 − 𝑇∞
= 𝑒
−√(
ℎ𝑃
𝐾𝐴
)𝑥
𝐥𝐧(𝑻 − 𝑻∞) = 𝒍𝒏(𝑻𝒔 − 𝑻∞) − √(
𝒉𝑷
𝑲𝑨
)𝒙
y = 0.0002x
0
0.01
0.02
0.03
0.04
0.05
0.06
0 50 100 150 200 250 300 350
x^2(m^2)
t(seconds)
A Graph of x^2 against time t(sec)
29. A graph of ln(𝑇 − 𝑇∞) against x gives an intercept 𝑙𝑛(𝑇𝑠 − 𝑇∞) from which 𝑇𝑠 can
be measured.
From experiment, using an aluminium rod of radius 2mm and using a
thermoconductivity value of 𝟐𝟑𝟖 𝑾
𝒎𝑲
⁄ , The heat transfer coefficient h of
aluminium was found to be 𝟑. 𝟎𝟓𝟓𝟐𝟓 𝑾
𝒎𝟐𝑲
⁄ .
From experiment a graph of temperature (℃) against distance 𝑥 looks as below
for an aluminium rod of radius 2mm in steady state:
The value of 𝑇𝑠 is lower than the value of the flame 𝑇𝑓 because of heat
convection at the beginning of the rod. i.e.
From the equation
0
50
100
150
200
250
300
350
400
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
Temperature
(C)
Distance (x)
A graph of temperature against distance x
30. 𝑇 − 𝑇∞
𝑇𝑠 − 𝑇∞
= 𝑒
−√(
ℎ𝑃
𝐾𝐴
)𝑥
Plotting a graph of ln(𝑇 − 𝑇∞) against x (excluding temperature at x=0) gives an
intercept 𝑙𝑛(𝑇𝑠 − 𝑇∞) from which 𝑇𝑠 can be measured but the value of 𝑇𝑠 got is
not the value of the temperature of the flame at 𝑥 = 0(𝑻𝒇).
THEORY
There is a relationship between 𝑇𝑠 and temperature of the flame(𝑻𝒇) at 𝑥 = 0.
First of all, we can postulate an existence of a flux at the beginning of the rod
independent of time i.e.,
𝑞̇|𝑥=0 = ℎ𝑓(𝑇𝑓 − 𝑇𝑠)
Where:
ℎ𝑓 = ℎ𝑒𝑎𝑡 𝑡𝑟𝑎𝑛𝑠𝑓𝑒𝑟 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑙𝑢𝑖𝑑 𝑎𝑡 𝑏𝑒𝑔𝑖𝑛𝑛𝑖𝑛𝑔 𝑜𝑓 𝑟𝑜𝑑
But we CAN’T equate this flux to
𝑞̇ = 𝑘
𝜕𝑇
𝜕𝑥
|𝑥=0
Because
𝑘
𝜕𝑇
𝜕𝑥
|𝑥=0 = 𝑘
(𝑇𝑠 − 𝑇∞)
𝛿
31. We have already derived 𝛿 and it is a function of time and using it to get 𝑇𝑠 will
cause 𝑇𝑠 to be a function of time yet a graph of 𝑥2
against time showed this is
not true since 𝑇𝑠 is constant independent of time.
We also can’t equate the above power (i.e., flux times area) to this power
ℎ𝑓𝜋𝑟2
(𝑇𝑓 − 𝑇𝑠) ≠ ℎ2𝜋𝑟 ∫ (𝑇 − 𝑇∞)𝑑𝑥
𝑙=∞
0
Because
ℎ2𝜋𝑟 ∫ (𝑇 − 𝑇∞)𝑑𝑥
𝑙=∞
0
= ℎ2𝜋𝑟𝛿(𝑇𝑠 − 𝑇∞)
The above would also bring back 𝛿 which is a function of time and this would
mean 𝑇𝑠 is a function of time which would contradict the observation of the
graph 𝑥2
against time to be a straight-line graph.
So, the only option we are left with is equating the fluxes at 𝒙 = 𝟎 below:
𝒉𝒇(𝑻𝒇 − 𝑻𝒔) = 𝒉(𝑻𝒔 − 𝑻∞)
The expression above will give us a temperature 𝑇𝑠 independent of time since all
the above factors don’t depend on time.
We can make 𝑇𝑠 the subject of the formula and get:
𝑻𝒔 =
𝒉𝒇𝑻𝒇 + 𝒉𝑻∞
𝒉 + 𝒉𝒇
From the above, it can be seen that 𝑇𝑠 is independent of time. From experiment,
it was found that 𝒉𝒇 = 𝟎. 𝟑𝟏𝟖𝟐 𝑾/(𝒎. 𝑲) independent of radius of the metal rod.
The temperature of the flame used was measured to be 379.5℃.
Therefore, the equation becomes:
𝑥2
= 2𝛼[ln (
𝑇𝑠 − 𝑇∞
𝑇 − 𝑇∞
)]2
× 𝑡 −
𝑎𝐾𝑟
ℎ
[ln (
𝑇𝑠 − 𝑇∞
𝑇 − 𝑇∞
)]2
𝒙𝟐
= 𝟐𝜶[𝐥𝐧 (
(
𝒉𝒇𝑻𝒇 + 𝒉𝑻∞
𝒉 + 𝒉𝒇
) − 𝑻∞
𝑻 − 𝑻∞
)]𝟐
× 𝒕 −
𝒂𝑲𝒓
𝒉
[𝐥𝐧 (
(
𝒉𝒇𝑻𝒇 + 𝒉𝑻∞
𝒉 + 𝒉𝒇
) − 𝑻∞
𝑻 − 𝑻∞
)]𝟐
32. How do we measure the heat transfer coefficient?
From experiment, using an aluminium rod of radius 2mm and using a
thermoconductivity value of 𝟐𝟑𝟖 𝑾
𝒎𝑲
⁄ , the heat transfer coefficient h of
aluminium was found to be 𝟑. 𝟎𝟓𝟓𝟐𝟓 𝑾
𝒎𝟐𝑲
⁄ .
From,
𝐥𝐧(𝑻 − 𝑻∞) = 𝒍𝒏(𝑻𝒔 − 𝑻∞) − √(
𝒉𝑷
𝑲𝑨
)𝒙
The gradient of A graph of ln(𝑇 − 𝑇∞) against x (excluding temperature at x=0)
gives √(
ℎ𝑃
𝐾𝐴
) as the gradient from which h can be measured.
h can also be got from Stefan’s law of cooling in natural convection that
reduces to the Newton’s law of cooling.
Stefan’s law of cooling in natural convection in a non-vacuum environment
states:
𝒅𝑸
𝒅𝒕
= (𝟏 + 𝑮)𝑨𝝈𝜺[𝑻𝟒
− 𝑻∞
𝟒
]
Where:
𝑮 = 𝒌𝑷𝒓𝒏
= 𝒆𝒙𝒑𝒆𝒓𝒊𝒎𝒆𝒏𝒕𝒂𝒍 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕
𝒏 = 𝒆𝒙𝒑𝒆𝒓𝒊𝒎𝒆𝒏𝒕𝒂𝒍 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕
𝜺 = 𝒆𝒎𝒊𝒔𝒔𝒊𝒗𝒊𝒕𝒚
𝝈 = 𝑺𝒕𝒆𝒇𝒂𝒏 𝑩𝒐𝒍𝒕𝒛𝒎𝒂𝒏𝒏 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕
𝑻 = 𝒕𝒆𝒎𝒑𝒆𝒓𝒂𝒕𝒖𝒓𝒆 𝒊𝒏 𝒌𝒆𝒍𝒗𝒊𝒏
𝑻∞ = 𝒓𝒐𝒐𝒎 𝒕𝒆𝒎𝒑𝒆𝒓𝒂𝒕𝒖𝒓𝒆 𝒊𝒏 𝒌𝒆𝒍𝒗𝒊𝒏
Where:
𝑃
𝑟 = 𝑃𝑟𝑎𝑛𝑑𝑡𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑢𝑟𝑟𝑜𝑢𝑛𝑑𝑖𝑛𝑔 𝑚𝑒𝑑𝑖𝑢𝑚 𝑎𝑡 𝑟𝑜𝑜𝑚 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒
The Prandtl number above is independent of temperature of the cooling body.
33. Considering
𝑇 = 𝑇∞ + ∆𝑇
𝑑𝑄
𝑑𝑡
= (1 + 𝐺)𝐴𝜎𝜀[(𝑇∞ + ∆𝑇)4
− 𝑇∞
4
]
Factorizing out 𝑇∞, we get
𝑑𝑄
𝑑𝑡
= (1 + 𝐺)𝐴𝜎𝜀[𝑇∞
4
(1 +
(𝑇 − 𝑇∞)
𝑇∞
)4
− 𝑇∞
4
]
It is known from Binomial expansion that:
(1 + 𝑥)𝑛
≈ 1 + 𝑛𝑥 𝑓𝑜𝑟 𝑥 ≪ 1
So:
(1 +
(𝑇 − 𝑇∞)
𝑇∞
)4
≈ 1 + 4
(𝑇 − 𝑇∞)
𝑇∞
= 1 + 4
∆𝑇
𝑇∞
= 𝑓𝑜𝑟
∆𝑇
𝑇∞
≪ 1
Simplifying, we get Newton’s law of cooling i.e.
𝒅𝑸
𝒅𝒕
= 𝟒(𝟏 + 𝑮)𝑨𝝈𝜺𝑻∞
𝟑 (𝑻 − 𝑻∞)
𝒅𝑸
𝒅𝒕
= 𝒉𝑨(𝑻 − 𝑻∞)
Where:
𝒉 = 𝟒(𝟏 + 𝑮)𝝈𝜺𝑻∞
𝟑
Substitute for the above parameters of aluminium and get h theoretically and
compare as got experimentally.
How do we deal with metal rods of finite length 𝒍 ?
34. The boundary and initial conditions are
𝑻 = 𝑻𝒔 𝒂𝒕 𝒙 = 𝟎
−𝒌
𝒅𝑻
𝒅𝒙
= 𝒉(𝑻 − 𝑻∞) 𝒂𝒕 𝒙 = 𝒍
𝑻 = 𝑻∞ 𝒂𝒕 𝒕 = 𝟎
Let us go about solving for the above boundary conditions
We start with a temperature profile below:
𝑇 − 𝑇∞ = (𝑇𝑠 − 𝑇∞)𝑒−
𝑥
𝛿
(1−
𝑥
𝑙
)
[
𝑥
𝑙
(𝑇1 − 𝑇∞)
(𝑇𝑠 − 𝑇∞)
+ (1 −
𝑥
𝑙
)]
Which says
𝑇 = 𝑇𝑠 𝑎𝑡 𝑥 = 0
𝑇 = 𝑇1 𝑎𝑡 𝑥 = 𝑙
𝑇 = 𝑇∞ 𝑎𝑡 𝑡 = 0
Provided 𝛿 = 0 𝑎𝑡 𝑡 = 0 , then the initial condition above is satisfied
35. we take the derivative
𝒅𝑻
𝒅𝒙
𝒂𝒕 𝒙 = 𝒍 and equate it to −
ℎ
𝑘
(𝑇1 − 𝑇∞) and get:
𝑑𝑇
𝑑𝑥
|𝑥=𝑙 = (
(𝑇1 − 𝑇∞)
𝑙
−
(𝑇𝑠 − 𝑇∞)
𝑙
+
(𝑇1 − 𝑇∞)
𝛿
)
𝑑𝑇
𝑑𝑥
|𝑥=𝑙 = −
ℎ
𝑘
(𝑇1 − 𝑇∞)
We equate the two and get
(
(𝑇1 − 𝑇∞)
𝑙
−
(𝑇𝑠 − 𝑇∞)
𝑙
+
(𝑇1 − 𝑇∞)
𝛿
) = −
ℎ
𝑘
(𝑇1 − 𝑇∞)
We finally get
(𝑇1 − 𝑇∞) = (𝑇𝑠 − 𝑇∞)(
𝛿𝑘
𝛿𝑘 + 𝑙𝑘 + ℎ𝑙𝛿
)
We substitute 𝑇1 − 𝑇∞ into the temperature profile and get
(𝑻 − 𝑻∞) = (𝑻𝒔 − 𝑻∞)𝒆−
𝒙
𝜹
(𝟏 −
𝒙
𝒍
)
[
𝒙
𝒍
(
𝜹𝒌
𝜹𝒌 + 𝒍𝒌 + 𝒉𝒍𝜹
) + (𝟏 −
𝒙
𝒍
)]
This the temperature profile that satisfies the boundary and initial conditions
below
𝑻 = 𝑻𝒔 𝒂𝒕 𝒙 = 𝟎
−𝒌
𝒅𝑻
𝒅𝒙
= 𝒉(𝑻 − 𝑻∞) 𝒂𝒕 𝒙 = 𝒍
𝑻 = 𝑻∞ 𝒂𝒕 𝒕 = 𝟎
Let us go ahead and solve for 𝛿
The governing equation is
𝛼
𝜕2
𝑇
𝜕𝑥2
−
ℎ𝑃
𝐴𝜌𝐶
(𝑇 − 𝑇∞) =
𝜕𝑇
𝜕𝑡
Let us change this equation into an integral as below:
𝛼 ∫ (
𝜕2
𝑇
𝜕𝑥2
) 𝑑𝑥
𝑙
0
−
ℎ𝑃
𝐴𝜌𝐶
∫ (𝑇 − 𝑇∞)𝑑𝑥
𝑙
0
=
𝜕
𝜕𝑡
∫ (𝑇)𝑑𝑥
𝑙
0
… … . . 𝑏)
38. 2ℎ𝑃
𝐴𝜌𝐶
𝑡 ≪ 1
𝑒
−ℎ𝑃
𝐴𝜌𝐶
𝑡
= 1 −
2ℎ𝑃
𝐴𝜌𝐶
𝑡
1 − 𝑒
−2ℎ𝑃
𝐴𝜌𝐶
𝑡
=
2ℎ𝑃
𝐴𝜌𝐶
𝑡
𝛿 = √2𝛼𝑡
We substitute for 𝛿 in the temperature profile.
(𝑻 − 𝑻∞) = (𝑻𝒔 − 𝑻∞)𝒆
−
𝒙
√𝟐𝜶𝒕
(𝟏 −
𝒙
𝒍
)
[
𝒙
𝒍
(
𝒌√𝟐𝜶𝒕
𝒌√𝟐𝜶𝒕 + 𝒍𝒌 + 𝒉𝒍√𝟐𝜶𝒕
) + (𝟏 −
𝒙
𝒍
)]
The above equation is observed for small times.
What is the flux at 𝑥 = 𝑙
From
−𝒌
𝒅𝑻
𝒅𝒙
= 𝒉(𝑻𝟏 − 𝑻∞) 𝒂𝒕 𝒙 = 𝒍
Substitute for (𝑻𝟏 − 𝑻∞) and get
𝑞̇|=𝑙 = ℎ(𝑇𝑠 − 𝑇∞)(
𝛿𝑘
𝛿𝑘 + 𝑙𝑘 + ℎ𝑙𝛿
)
You notice that at 𝑙 = 0
𝑞̇|𝑙=0 = ℎ(𝑇𝑠 − 𝑇∞)
And at 𝑙 = ∞
𝑞̇|𝑙=∞ = 0
Which is true.
What happens when the length is big or tends to infinity?
(𝑻 − 𝑻∞) = (𝑻𝒔 − 𝑻∞)𝒆−
𝒙
𝜹
(𝟏 −
𝒙
𝒍
)
[
𝒙
𝒍
(
𝜹𝒌
𝜹𝒌 + 𝒍𝒌 + 𝒉𝑳𝒍𝜹
) + (𝟏 −
𝒙
𝒍
)]
Becomes
(𝑻 − 𝑻∞) = (𝑻𝒔 − 𝑻∞)𝒆−
𝒙
𝜹
39. (𝑻 − 𝑻∞) = (𝑻𝒔 − 𝑻∞)𝒆
−
𝒙
√𝑲𝑨
𝒉𝑷
(𝟏−𝒆
−𝟐𝒉𝑷
𝑨𝝆𝑪
𝒕
)
Which is what we got before.
40. WHAT HAPPENS WHEN THE INITIAL TEMPERATURE IS
A FUNCTION OF X?
The governing equation is
𝛼
𝜕2
𝑇
𝜕𝑥2
=
𝜕𝑇
𝜕𝑡
BCs
𝑻 = 𝑻𝒔 𝒂𝒕 𝒙 = 𝟎 𝟎 < 𝒕 < ∞
𝑻 = 𝑻𝒔 𝒂𝒕 𝒙 = 𝒍 𝟎 < 𝒕 < ∞
IC
𝑻 = ∅(𝒙) 𝒂𝒕 𝒕 = 𝟎 𝟎 ≤ 𝒙 ≤ 𝒍
We assume an exponential temperature profile that satisfies the boundary
conditions:
𝑇 − ∅
𝑇𝑠 − ∅
= 𝑒
−𝑥
𝛿
(1−
𝑥
𝑙
)
The PDE becomes an integral equation given by:
𝛼 ∫ (
𝜕2
𝑇
𝜕𝑥2
) 𝑑𝑥
𝑙
0
=
𝜕
𝜕𝑡
∫ 𝑇𝑑𝑥
𝑙
0
Let us give an example say
∅ = 𝑥
We make T the subject of the formula and get
𝑇 = ∅ + 𝑇𝑠𝑒
−𝑥
𝛿
(1−
𝑥
𝑙
)
− ∅𝑒
−𝑥
𝛿
(1−
𝑥
𝑙
)
∅ = 𝑥
𝑇 = 𝑥 + 𝑇𝑠𝑒
−𝑥
𝛿
(1−
𝑥
𝑙
)
− 𝑥𝑒
−𝑥
𝛿
(1−
𝑥
𝑙
)
∫ (
𝜕2
𝑇
𝜕𝑥2
) 𝑑𝑥
𝑙
0
= [
𝜕𝑇
𝜕𝑥
]
𝑙
0
We go ahead and solve for 𝛿.
41. Using this integral analytical method, we can also go ahead and solve PDES of
the form below:
𝜕𝑇
𝜕𝑡
= 𝛼
𝜕2
𝑇
𝜕𝑥2
+ 𝑓(𝑥)
42. HOW DO WE DEAL WITH CYLINDRICAL CO-ORDINATES
FOR AN INFINITE RADIUS CYLINDER?
We know that for an insulated cylinder where there is no heat loss by
convection from the sides, the governing PDE equation is
𝛼 [
𝜕2
𝑇
𝜕𝑟2
+
1
𝑟
𝜕𝑇
𝜕𝑟
] =
𝜕𝑇
𝜕𝑡
The boundary conditions are
𝑇 = 𝑇𝑠 𝑎𝑡 𝑟 = 𝑟1
𝑇 = 𝑇∞ 𝑎𝑡 𝑟 = ∞
The initial condition is:
𝑇 = 𝑇∞ 𝑎𝑡 𝑡 = 0
The temperature profile that satisfies the conditions above is
𝑇 − 𝑇∞
𝑇𝑠 − 𝑇∞
= 𝑒
−(𝑟−𝑟1)
𝛿
We transform the PDE into an integral equation
𝛼 [
𝜕2
𝑇
𝜕𝑟2
+
1
𝑟
𝜕𝑇
𝜕𝑟
] =
𝜕𝑇
𝜕𝑡
We take integrals from 𝑟1 to 𝑟 = 𝑅 = ∞
𝛼 ∫ (
𝜕2
𝑇
𝜕𝑟2
) 𝑑𝑟
𝑅
𝑟1
+ 𝛼 ∫ [
1
𝑟
(
𝜕𝑇
𝜕𝑟
)]𝑑𝑟
𝑅
𝑟1
=
𝜕
𝜕𝑡
∫ 𝑇𝑑𝑟
𝑅
𝑟1
We then go ahead to solve and find 𝛿 as before.
∫ (
𝜕2
𝑇
𝜕𝑟2
) 𝑑𝑟
𝑅
𝑟1
= [
𝜕𝑇
𝜕𝑟
]
𝑅
𝑟1
= −
𝑇𝑠 − 𝑇∞
𝛿
[𝑒
−(𝑟−𝑟1)
𝛿 ]
𝑅 = ∞
𝑟1
=
(𝑇𝑠 − 𝑇∞)
𝛿
𝜕𝑇
𝜕𝑟
= −
𝑇𝑠 − 𝑇∞
𝛿
𝑒
−(𝑟−𝑟1)
𝛿
∫ [
1
𝑟
(
𝜕𝑇
𝜕𝑟
)] 𝑑𝑟
𝑅
𝑟1
= −
(𝑇𝑠 − 𝑇∞)
𝛿
∫
1
𝑟
𝑒
−(𝑟−𝑟1)
𝛿 𝑑𝑟
𝑅
𝑟1
∫
1
𝑟
𝑒
−(𝑟−𝑟1)
𝛿 𝑑𝑟
𝑅
𝑟1
= 𝑢𝑣 − ∫ 𝑣
𝑑𝑢
𝑑𝑟
𝑑𝑟
44. 𝜕
𝜕𝑡
∫ 𝑇𝑑𝑟
𝑅
𝑟1
=
𝑑𝛿
𝑑𝑡
(𝑇𝑠 − 𝑇∞)
substituting all the above in the integral equation, we get
𝛼 ∫ (
𝜕2
𝑇
𝜕𝑟2
) 𝑑𝑟
𝑅
𝑟1
+ 𝛼 ∫ [
1
𝑟
(
𝜕𝑇
𝜕𝑟
)]𝑑𝑟
𝑅
𝑟1
=
𝜕
𝜕𝑡
∫ 𝑇𝑑𝑟
𝑅
𝑟1
𝛼
(𝑇𝑠 − 𝑇∞)
𝛿
−
𝛼(𝑇𝑠 − 𝑇∞)
𝛿 + 𝑟1
=
𝑑𝛿
𝑑𝑡
(𝑇𝑠 − 𝑇∞)
We go ahead and solve for 𝛿
𝛼
𝛿
−
𝛼
𝛿 + 𝑟1
=
𝑑𝛿
𝑑𝑡
The boundary conditions are:
𝛿 = 0 𝑎𝑡 𝑡 = 0
We solve for 𝛿 and get an algebraic cubic equation
𝟐𝜹𝟑
+ 𝟑𝒓𝟏𝜹𝟐
− 𝟔𝒓𝟏𝜶𝒕 = 𝟎
From which 𝛿 can be got.
We can also go ahead and look at situations where there is natural convection
and other situations where the radius r is finite and not infinite.