# James Vere BeckMichigan State University | MSU · Department of Mechanical Engineering

James Vere Beck

Ph.D.

## About

206

Publications

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9,518

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Citations since 2016

Introduction

Additional affiliations

September 1963 - present

## Publications

Publications (206)

The primary use of analytical solutions in the area of thermal conduction problems is for verification
purposes, comparing the calculated temperatures and heat flux values to the results from numerical codes. The
contribution from the analytical solutions can be especially significant where large temperature gradients are found.
This is because...

This is the 33rd in the series of National and International meetings on Inverse Problems that were initiated at Michigan State University, East Lansing, MI, USA, in 1988. The 2022 Edition is the 10th International Conference.
Due to Coronavirus emergency and to protect the health and safety to
all our participants, the 10th Int. Conf. on Inverse...

Analytical solutions for thermal conduction problems are extremely important, particularly for verification of numerical codes. Temperatures and heat fluxes can be calculated very precisely, normally to eight or ten significant figures, even in situations involving large temperature gradients. It can be convenient to have a general analytical solut...

A two-dimensional transient thermal conduction problem is examined and numerical solutions to the problem generated by ANSYS and Matlab, employing the finite element (FE) method, are compared against an 'intrinsically' verified analytical solution. Various grid densities and time-step combinations are used in the numerical solutions, including some...

An analytical model using Green's functions for partial external heating of a pipe is developed, which results in an exact mathematical solution for the radial and axial temperature distribution in the pipe wall. Partial heating consists of a constant heat flux function imposed over a small section of the exterior of a pipe and for a limited time d...

A two-dimensional transient thermal conduction problem is examined and numerical solutions to the problem generated by ANSYS and Matlab, employing the finite element method, are compared against an analytical solution. Various different grid densities and time-step combinations are used in the numerical solutions, including some as recommended by d...

The development of a generalized solution is presented for a three-dimensional transient heat conduction problem in a rectangular parallelepiped. To make the method as general as possible, one face of the body is subjected to a nonhomogeneous boundary condition over part of the surface. The solution accommodates three kinds of boundary conditions:...

A novel formulation for numerical solution of heat conduction problems using superposition of exact solutions (SES) to represent temperature on sub-elements of a region is described and demonstrated. A simple 1-D linear problem is used to describe the method and highlight potential benefits; however, extensions to higher geometric dimensions and li...

A generalized solution for a two-dimensional (2D) transient heat conduction problem
with a partial-heating boundary condition in rectangular coordinates is developed. The
solution accommodates three kinds of boundary conditions: prescribed temperature, prescribed
heat flux and convective. Also, the possibility of combining prescribed heat flux
and...

A desirable feature of any parameter estimation method is to obtain as much information
as possible with one experiment. However, achieving multiple objectives with one experiment
is often not possible. In the field of thermal parameter estimation, a determination
of thermal conductivity, volumetric heat capacity, heat addition rate, surface emissi...

A generalized solution for a two-dimensional transient heat conduction problem with a partialheating
boundary condition in rectangular coordinates is developed. The solution accommodates
three kinds of boundary conditions: prescribed temperature, prescribed heat flux and convective.
Also, the possibility of combining prescribed heat flux and convec...

In the design of transient thermal experiments for estimating the thermal conductivity, k, and volumetric heat capacity, C, the scaled sensitivity coefficients are utilized. These coefficients should be large and uncorrelated. In the current paper it is proven that they are the largest when the heated surface temperature of the sample is held at th...

This paper provides a solution for two-dimensional (2D) heating over a rectangular
region on a homogeneous plate. It has application to verification of numerical conduction
codes as well as direct application for heating and cooling of electronic equipment. Additionally,
it can be applied as a direct solution for the inverse heat conduction problem...

Numerical codes are important in providing solutions to partial differential equations in many areas, such as the heat transfer problem. However, verification of these codes is critical. A methodology is presented in this work as an intrinsic verification method (IVM) to the solution to the partial differential equation. Derivation of the dimension...

A desirable feature of any parameter estimation method is to obtain as much information as possible with one experiment. However, achieving multiple objectives with one experiment is often not possible. In the field of thermal parameter estimation, a determination of thermal conductivity, volumetric heat capacity, heat addition rate, surface emissi...

Estimation of thermal properties or diffusion properties from transient data requires that a model is available that is physically meaningful and suitably precise. The model must also produce numerical values rapidly enough to accommodate iterative regression, inverse methods, or other estimation procedures during which the model is evaluated again...

An exact solution is presented for two-dimensional transient heat conduction in a rectangular plate heated at y = 0 from x = 0 to x = L1 and insulated over the other edges. This problem does not have a steady-state solution, but does have a quasi-steady solution. Because of this, Green's functions are used to determine the exact solution. The solut...

Heat transfer in solids provides an opportunity for students to learn of several boundary conditions: the first kind for specified temperature, the second kind for specified heat flux, and the third kind for specified convection. In this paper we explore the relationship among these types of boundary conditions in steady heat transfer. Specifically...

This paper provides a solution for two-dimensional heating over a rectangular region on a homogeneous plate. It has application to verification of numerical conduction codes as well as direct application for heating and cooling of electronic equipment. Additionally, it can be applied as a direct solution for the inverse heat conduction problem, mos...

Estimating thermal properties for thick or solid foods at temperatures greater than 100 °C is challenging for two reasons: the long time needed to reach a constant temperature, and the pressure needed to be maintained in the sealed container. An instrument (TPCell) was developed based on a rapid non-isothermal method to estimate the temperature-dep...

The time duration for processes involving transient thermal diffusion can be a critical piece of information related to thermal processes in engineering applications. Analytical solutions must be used to calculate these types of time durations because the boundary conditions in such cases can be effectively like semi-infinite conditions. This resea...

The inverse heat conduction problem is the estimation of the time and/or space dependence of the surface heat flux or temperature utilizing interior temperature measurements at discrete times and/or locations. This problem is ill-posed since it is very sensitive to omnipresent measurement errors. Many solution methods have been proposed including e...

Heat flux measurement is essential in several industrial applications. While direct measurement of heat flux is not a simple task and sometimes is not even possible, measuring temperature is much easier and viable in a variety of applications. Heat flux estimation using temperature measurement data requires solving inverse heat conduction problems...

Applied computer solutions for conductive heat transfer are a critical component in any modern undergraduate heat transfer course. This need has been addressed in many ways through various textbook exercises and software packages. The present work involves a catalog of analytical solutions organized with a numbering system that describes the bounda...

Accurate measurement of heat flux is important in numerous industrial applications. Heat flux estimation using temperature measurement data requires solving inverse heat conduction problems (IHCP's). In the present paper, a real-time solution for two-dimensional inverse heat conduction problem is presented. It is assumed that multiple unknown heat...

Real-time measurement of heat flux is an important challenge for several industrial applications, including furnace control. For efficient operation of high-temperature process furnaces, accurate and stable temperature measurements are needed. Directional Flame Thermometer (DFT) offers the ability to use both temperature and heat flux measurements...

The time duration for processes involving transient thermal diffusion can be a critical piece of information related to thermal processes in engineering applications. Analytical solutions must be used to calculate these types of time durations since the boundary conditions in such cases can be effectively like semi-infinite conditions. This researc...

This paper presents a solution for the inverse heat conduction problem (IHCP) in a multi-layer medium based on solutions from individual layers separately. The approach allows for inclusion of known contact resistances between the layers. The temperature histories are assumed known at two points on the inner layer and the heat transfer rate at the...

The inverse heat conduction problem (IHCP) involves estimation of a surface heat flux from transient temperature measurements inside a heat conducting body. Commonly an insulated remote boundary or one with a known heat transfer coefficient is modeled. However, in many practical applications, the precise thermal condition at the remote boundary is...

The real-time measurement of heat flux is an important challenge for several industrial applications including furnace control. For efficient operation of high-temperature process furnaces, accurate and stable temperature measurements are needed. Directional Flame Thermometers (DFTs) offer the ability to use both temperature and heat flux measureme...

A new solution is presented for one-dimensional heat conduction with boundary heat flux increasing as fourth power of time. Green's functions (GFs) are used and resulting slowly-converging summations in the solution are replaced with equivalent closed-form expressions. Solutions to time power-law heat flux variations that stop suddenly after a spec...

There are many applications for problems involving thermal conduction in two-dimensional cylindrical objects. Experiments involving thermal parameter estimation are a prime example, including cylindrical objects suddenly placed in hot or cold environments. In a parameter estimation application, the direct solution must be run iteratively in order t...

The thermal wave effect occurs during rapid heating of pure dielectric materials. It also serves as a limiting solution for rapid heating of other materials. The numerical and series solutions for this type of problems are well documented in the literature. This work uses a finite series solution at small time to prepare a Green's function (GF) sol...

Tikhonov regularization for the inverse heat conduction problem (IHCP) is considered a "whole domain" or "batch" method, meaning that observations are needed over the entire time domain of interest, and that calculations must be performed all-at-once in a batch. This paper examines the structure of the Tikhonov regularization problem and concludes...

This paper is intended to provide very accurate analytical solutions modeling transient heat conduction processes in 2D Cartesian finite bodies for small values of the time. Analysis of diffusion of thermal deviation effects indicates that, when the space and time coordinates satisfy a criterion developed in the paper, the simple transient 1D semi-...

This paper is intended to provide very accurate analytical solutions modeling
transient heat conduction processes in 2D Cartesian finite bodies for small values of the time.
Analysis of diffusion of thermal deviation effects indicates that, when the space and time
coordinates satisfy a criterion developed in the paper, the simple transient 1D semi-...

A transient, multi-dimensional, heat conduction problem can be solved using analytical (exact and
approximate) and numerical methods [1, 2]. They are the first stage of solution procedures for solving the
inverse heat conduction problems (IHCPs) [3]. Among them, the numerical approximate form of the Green’s
function equation based on a heat-flux fo...

A numerical approximation of the Green’s function equation based on a heat-flux formulation is given. It is derived by assuming as a functional form of the surface heat flux a stepwise variation with space and time. The obtained approximation is very important in investigation of the inverse heat conduction problems (IHCPs) because it gives a conve...

The geothermal or ground-source heat pump (GHP) has been shown to be a very efficient method of providing heating and cooling for buildings. GHPs exchange (reject or extract) heat with the earth by way of circulating water, rather than by use of circulating outdoor air, as with an air-source heat pump. The temperature of water entering a GHP is gen...

An exact solution method is given for 3D heating over a rectangular region on a homogeneous plate. It has application to laser heating and cooling of electronic equipment. Based on Green’s functions and new summation identities, significant improvements are given over the separation of variables method. Machine accuracy is possible, which can be us...

In mixed boundary value (MBV) problems, the nature of the boundary condition can change along a particular boundary (finite, semi-infinite or infinite in length), say from a Dirichlet condition to a Neumann condition. Most MBV problems are solved using classical techniques such as separation of variables (domain of limited extent) or transform meth...

This paper presents the state-variable modal decomposition of transient temperature
ensemble data of into modal components to estimate time constants and characteristic shapes of a heated rod.

The objective of this presentation is the development of a generalized steady-state Green’s function solution to study the temperature field in moving bodies. This type of solution permits the inclusion of different non-homogeneous boundary conditions, volumetric heat sources, and possible position-dependent thermophysical properties. Although the...

One of the recommendations that came from the NIST investigation of the World Trade Center disaster was the need for quantitative heat flux measurements in larger scale fire safety tests. These heat flux data are needed to support the development of engineering models to predict the performance of fire protection materials and systems. Current stan...

A technique is presented for the uncertainty analysis of the linear Inverse Heat Conduction Problem (IHCP) of estimating heat flux from interior temperature measurements. The selected IHCP algorithm is described. The uncertainty in thermal properties and temperature measurements is considered. A propagation of variance equation is used for the unce...

A two-dimensional heat conduction problem in Cartesian coordinates subject to a periodic-in-space boundary condition is analyzed by the Green’s functions approach. It is pointed out that when the frequency of the spatial periodic heating equates one of the natural frequencies (eigenvalues) of the system, the solution of the 2D heat conduction probl...

This paper describes the transport of thermal energy within a small distance after an abrupt wall temperature change in a
circular duct. In general, the axial conduction becomes significant when the Peclet number is small. The results indicate
that the inclusion of axial conduction in the fluid substantially increases the wall heat flux at near the...

Steady state conduction of heat from a stationary wall to a medium moving at a uniform velocity is the subject herein. This medium can be a solid or a fluid moving at a constant velocity. The surface of this medium is insulated until a change in the surface heat flux occurs. The determination of temperature field is the main objective herein. The r...

Optimal experimental design for nonlinear models is important to enhance accuracy of parameter estimates. Analysis for optimal design was conducted for the thermal processing of grape pomace (moisture content 42% wb) at constant retort temperarture of 126.7oC. The optimum heating time is proposed that minimizes the hypervolume of the confidence reg...

Exact series solutions for the computation of temperature in parallel plate channels and circular passages are well known. The inclusion of the contribution of axial conduction leads to a set of modified Graetz type problems for these fluid passages. The emphasis of this paper is the study of the asymptotic variations of wall heat flux values adjac...

This paper analyzes the diffusion of thermal disturbances in heat-conducting two-dimensional rectangular bodies through characteristic times, such as penetration and deviation times, denoting their effects within a certain order of magnitude. A single basic criterion governing the above diffusion is derived thanks to the similarity of the findings....

The paper develops a procedure for the theoretical-experimental calculation of the solar absorption coefficient of metallic layers under the action of a short flash of concentrated solar energy. The knowledge of this coefficient is relevant when the solar thermal processing is used to make a quench in a thin superficial layer of metallic slabs. The...

This paper considers the steady state conduction of heat from a wall to a fluid moving at a uniform velocity. The wall is heated by a step change in temperature. Although this appears to be a classical heat conduction problem, its application to various convective heat transfer problems is new. The mathematical procedure leads to the computation of...

Steady-state components of heat conduction solutions may have very slowly convergent series for temperatures and non-convergent heat fluxes for temperature boundary conditions. Previous papers have proposed methods to remove these convergence problems. However, even more effective procedures based oil insights of Morse and Feshbach are given herein...

Transient temperature solutions in plates are derived for heating conditions varying as time to an integer power at a surface. The powers include 0, 1, 2 and 3; the last of which is useful for cubic splines. Boundary conditions of the first, second and third kinds are treated at both surfaces with the non-homogeneity at x=0. As the power increases...

The study of heat transfer in the entrance region of ducts with different cross-sections is important in engineering practice. This paper considers laminar, hydrodynamically fully developed flow in the thermal entrance regions of rectangular passages, emphasizing heat transfer aspects. By having a prescribed heating or cooling rate and considering...

Estimation of the heat flux for a quenched solid stainless steel sphere is demonstrated using a filter formulation. Even though the values of the thermal properties significantly change during quenching, nearly as accurate heat flux estimates as with temperature-variable analyses are possible using the proposed filter method. The filter algorithm i...

This paper presents an accurate methodology to determine heat transfer coefficients near the thermal entrance region of ducts. These include parallel plate channels, circular pipes and rectangular passages. The solution technique uses a classical Airy differential equation when the thermal penetration is small. The validation and verification of th...

Heat conduction in a rectangular parallelepiped that is in steady motion relative to a fluid is studied in this paper. The
governing equation consists of the standard heat equation plus lower-order derivative terms with the space variables that
represent the effects of the solid flow. The presence of the first-order-derivative terms with the space...

This paper addresses exact, transient heat-conduction solutions in two-dimensional rectangles heated at a boundary. The standard method of separation of variables (SOV) solution has two parts, steady-state (or quasi-steady) and complementary transient. The steady-state component frequently converges slowly at the heated surface, which is usually th...

The analytical solution for the problem of transient thermal conduction with solid body movement is developed for a parallelepiped with convective boundary conditions. An effective transformation scheme is used to eliminate the flow terms. The solution uses Green’s functions containing convolution-type integrals, which involve integration over a du...

In engineering applications, it is possible to heat up a solid body with constant input of thermal energy. Two different limiting cases are recognized, one maintaining a spatially constant surface temperature and the other having a constant wall heat flux. For geometries such as parallel plates and circular rods, there are direct solutions. However...

Verification of the codes that provide numerical heat transfer solutions obtained by finite difference and other methods is important. One way to verify these solutions is to compare the values with exact solutions. However, these exact solutions should also be verified. Fortunately, intrinsic verification methods are possible. Intrinsic verificati...

One method for verification of numerical solutions in heat transfer uses exact solutions for multi-dimensional parallelepipeds. The usual solution of these problems utilizes the method of separation of variables for both the steady and transient parts of the solution; however, this method for the steady-state part often produces solutions that conv...

The analytical solution for the problem of transient thermal conduction with solid body movement is developed for an orthotropic parallelepiped. Transformations are used to eliminate the flow terms and the orthotropic dependence. The solution uses two types of Green's functions: one coming from the Laplace transform method and the other from the me...

A boundary-value problem for steady-state heat conduction in a three-dimensional, two-layered composite is studied. The method of Green's function is used in the study. Green's functions are constructed as double sums in terms of eigenfunctions in two of the three directions. The eigenfunctions in the direction orthogonal to the layers are unconven...

Small time approximations are obtained for Green's functions in one-dimensional heat conduction problems with convective boundaries. The method of images similar to those for cases with Dirichlet or Neumann boundaries is used. Multiple reflections in this method generate infinite series representations for the Green's functions whose general terms...

The mathematical formulation of the steady-state temperature field in multi-dimensional and multi-layer bodies is presented. The numerical examples are for two-layer bodies and they include boundary conditions of the first, second, and third kind. This study includes tables to assist the selection of eigenfunctions and computation of the eigenvalue...

Verification solutions for the problem of combined thermal conduction and uniform flow are developed. Expressions and numerical values are given for transient one- and three-dimensional problems with temperature boundary conditions. Results are given for both the temperature and heat flux. Solutions for three sets of boundary conditions for the ste...

An initial-boundary value problem for transient heat conduction in a rectangular parallelepiped is studied. Solutions for the temperature and heat flux are represented as integrals involving the Green's function (GF), the initial and boundary data, and volumetric energy generation. Use of the usual GF obtained by separation of variables leads to sl...

An analytical solution is provided to the nonlinear diffusion equation, with the thermal conductivity given as a linear function of temperature. The derivation of the solution, and implications of it, are presented. The boundary and initial conditions associated with the solution provide applicability to specific cases. The solution is useful for v...

This article describes the development of accurate solutions for transient three-dimensional conductive heat transfer in Cartesian coordinates for a parallelepiped which is homogeneous and has constant thermal properties. The intended use of these solutions is for verification of numerical computer programs which are used for solving transient heat...

Mathematical steps leading to computation of the temperature field in multi-dimensional, multi-layer bodies are described and numerical results for two-layer bodies are presented. The presentations include boundary conditions of the first, second, and third kind. Included in this paper is a table to assist in computing eigenvalues. Also, modificati...

This paper reports the evaluation of a spectral technique for estimating thermophysical properties. It demonstrates that one can construct a virtual quasi-steady periodic experiment from a limited but properly selected set of transient non-periodic data. In the spectral domain, the phase angles of the responses at different locations relative to a...

Efficient algorithms for computing eigenvalues for heat conduction problems in Cartesian and spherical coordinates are given. Explicit approximate relations are presented that generally provide accurate results. When these approximate relations are followed by a high-order Newton root-finding iteration, a high degree of accuracy can be realized. It...

The condition of the final set of equations in any parameter estimating method is a critical feature and can often determine the success or failure of the method used. This aspect of parameter estimation becomes especially critical when sensitivity coefficients for parameters being estimated are correlated, which is a characteristic of ill-posed pr...

Parameter estimation techniques are applied to estimate temperature-dependent thermal properties from a series of transient experiments. Several experiments with one- and two-dimensional heat flow that cover a range from room temperature to 500 d