The distribution of temperature along a thin rod electrically heated in vacuo . V. Time lag

1955 ◽  
Vol 227 (1169) ◽  
pp. 141-154 ◽  
Keyword(s):  
Relaxation Time ◽  
Temperature Distribution ◽  
Fourier Expansion ◽  
Time Lag ◽  
Fourier Method ◽  
Effective Length ◽  
Close Approximation ◽  
Central Temperature ◽  
Actual Distribution ◽  
Special Case

On the basis of the expressions obtained in parts I and II of this series for the distribution of temperature in the steady state along a filament electrically heated in vacuo , the growth of temperature accompanying a small increase in the heating current is investigated in the present part. Over a considerable region about the centre of the filament, which is the region of practical interest, it is found that to a close approximation the growth of temperature can be completely represented by a simple exponential law involving a single relaxation time, whose magnitude is readily calculated. This method of investigating the time lag, which is general and applicable to any filament, is compared with the well-known method of Fourier expansion developed by Straneo for the special case where the temperature everywhere in the filament is only slightly higher than the room temperature, and hence the loss by radiation conforms to Newton’s law of cooling. Each of the Fourier terms is assigned in his method a separate relaxation time that will make the term separately satisfy the differential equation and the boundary conditions. In principle the Fourier method also should be applicable to any filament. But the actual temperature distribution is in general too complicated for an analytical Fourier expansion. In the special case treated by Straneo the temperature distribution over practically the whole length of the filament is parabolic. The actual distribution near the centre of any filament is also known to be parabolic. Hence a comparison of the results obtained by the two methods in the above special case suggests a convenient adaptation of the Fourier method also to the calculation of the time lag near the centre of any filament. The adaptation lies essentially in the use of a certain effective length to determine the period of the Fourier expansion, instead of the actual length generally used. The magnitude of this length is obtained from the results of the present investigation. The distinction between the two lengths is not significant in the special case treated by Straneo, but it is in other cases. Though the occurrence of a single effective relaxation time is not directly obvious from the Fourier expansion, it is shown to follow from it as a close approximation. This result is convenient for practical application. For a given central temperature the relaxation time is found to vary inversely as the ratio of the surface to the volume, and is therefore smaller for a ribbon filament than for one of circular cross-section, as observed by Prescott & Morrison. For a given central temperature and length, the ribbon filament is found to approximate closer than one of circular section, to an infinitely long one. The variation of the relaxation time near the centre with the length of the filament is investigated in some detail.

Virtual Population Analysis (VPA) Equations for Nonhomogeneous Populations, and a Family of Approximations Including Improvements on Pope's Cohort Analysis

1986 ◽  
Vol 43 (12) ◽  
pp. 2406-2409 ◽  
Author(s):  
Alec D. MacCall
Keyword(s):  
Mortality Rate ◽  
Population Analysis ◽  
Seasonal Pattern ◽  
Cohort Analysis ◽  
Natural Mortality ◽  
Close Approximation ◽  
Virtual Population Analysis ◽  
Virtual Population ◽  
Special Case

A set of "backward" virtual population analysis (VPA) equations relates catch (Ct) from continuous fishing between times t and t + 1 to population n size (Nt, Nt+1) when a portion of the stock is unavailable to fishing. The usual VPA equations become a special case where the entire stock is available (i.e. the stock is homogeneous). A close approximation to the VPA equations is Nt = Nt+1 exp(M) + CtM/(1 − exp(−M)), which has properties similar to Pope's "cohort analysis" and is somewhat more accurate in the case of a continuous fishery, especially if the natural mortality rate (M) is large. Much closer simple approximations are possible if the seasonal pattern of catches is known.

scholarly journals Transmissibility of epidemic diseases caused by delay with local proportional fractional derivative

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Abdullah Khamis Alzahrani ◽  
Oyoon Abdul Razzaq ◽  
Najeeb Alam Khan ◽  
Ali Saleh Alshomrani ◽  
Malik Zaka Ullah
Keyword(s):  
Biological Sciences ◽  
Fractional Derivative ◽  
Dynamic Assessment ◽  
Time Lag ◽  
Vital Role ◽  
Rate Of Change ◽  
Contagious Diseases ◽  
The Individual ◽  
Special Case ◽  
Epidemic Diseases

AbstractEpidemiological models have been playing a vital role in different areas of biological sciences for the analysis of various contagious diseases. Transmissibility of virulent diseases is being portrayed in the literature through different compartments such as susceptible, infected, recovered (SIR), susceptible, infected, recovered, susceptible (SIRS) or susceptible, exposed, infected, recovered (SEIR), etc. The novelty in this endeavor is the addition of compartments of latency and treatment with vaccination, so the system is designated as susceptible, vaccinated, exposed, latent, infected, treatment, and recovered (SVELITR). The contact of a susceptible individual to an infective individual firstly makes the individual exposed, latent, and then completely infection carrier. Innovatively, the assumption that exposed, latent, and infected individuals enter the treatment compartment at different rates after a time lag is also deliberated through the existence of time delay. The rate of change and constant solutions of each compartment are studied with incorporation of a special case of proportional fractional derivative (PFD). In addition, existence and uniqueness of the system are also comprehensively elaborated. Moreover, novel dynamic assessment of the system is carried out in context with the fractional order index. Succinctly, the manuscript accomplishes cyclic epidemiological behavior of the infectious disease due to the delay in treatment of the infected individuals.

On the solution of general thermo-magnetohydrodynamic elasticoviscous flow

1999 ◽  
Vol 77 (4) ◽  
pp. 279-297 ◽  
Author(s):  
K A Helmy
Keyword(s):  
Magnetic Field ◽  
Relaxation Time ◽  
Approximate Solution ◽  
Temperature Distribution ◽  
Oscillatory Flow ◽  
Energy Equation ◽  
Magnetic Parameter ◽  
Porous Plate ◽  
Elastic Parameter ◽  
Physical Parameters

Exact solutions of an oscillatory flow of an incompressible elasticoviscous conducting fluid with variable suction, in the presence of a constant magnetic field, applied perpendicular to the moving porous plate has been obtained. For small values of the elastic parameter k, the approximate solution of the problem has also been obtained.The temperature distribution is evaluated by solving the energy equation, taking into account the effect of the relaxation time. The effects of different physical parameters such as elastic parameter k, Prandtl number Pr, suction parameterA, magnetic parameter M, and the relaxation time τ on the velocity and temperature are discussed. Different values of physical parameters are tabulated and discussed numerically and graphically. PACS Nos.: 47.50, 47.65, 68.10.E

Nonsteady-State Particle Flow Under Gravity—An Extension of the Stochastic Theory

1974 ◽  
Vol 41 (4) ◽  
pp. 867-872 ◽  
Author(s):  
W. W. Mullins
Keyword(s):  
Particle Density ◽  
Time Lag ◽  
Normal Component ◽  
Stochastic Theory ◽  
Particle Flow ◽  
Lag Effects ◽  
Flow Disturbances ◽  
Nonsteady State ◽  
Random Flight ◽  
Special Case

The steady-state (ss) stochastic theory of convergent, cohesionless particle flow under gravity toward an orifice in the floor of a semi-infinite bed, based on the statistics of random flight and assuming instantaneous propagation of flow disturbances throughout the bed, is extended to nonsteady-state flow and time lag effects. The new theory, of which the ss theory is a special case, assumes flow to be restricted to an expanding zone, surmounting the orifice (opened at t = 0), of particle density ρss, separated from the rest of the bed of the original particle density ρ0 = ρss + Δρ (Δρ > 0) by a boundary whose elements advance with a velocity vn = −(1/Δρ)Jn where Jn is the normal component of the particle flux on the inside of the boundary due to flow (assumed to be ss) within the zone. Detailed equations describing the flow zone boundary as a function of time and the flow within the zone are developed; the equations depend on two material parameters (Δρ/ρss, and α of ss theory) and on the quantity of material drained out. Corrections are derived for the analysis of the z2 and z3/2 plots of layer data previously made on the basis of the ss theory. A comparison of the new predictions with one piece of flow data shows the theory capable of accounting for lag effects and for details of the flow pattern in that case. Values of Δρ/ρss and α are deduced, the latter being the order of the particle size in conformity to the expectations of the statistical theory.

scholarly journals The Effect of Relaxation Time on the Heat Transfer and Temperature Distribution in Tissues

2011 ◽  
Vol 01 (06) ◽  
pp. 283-287 ◽  
Author(s):  
Mohamed I. A. Othman ◽  
Mohamed Galal Sayed Ali ◽  
Roushdi Mohamed Farouk
Keyword(s):  
Heat Transfer ◽  
Relaxation Time ◽  
Temperature Distribution

scholarly journals Experimental Research on the Hydration Heat Temperature Field of Hollow Concrete Piers

2014 ◽  
Vol 8 (1) ◽  
pp. 482-487 ◽  
Author(s):  
Jun Peng ◽  
Chunping Tang ◽  
Liangliang Zhang ◽  
Ayad Thabit Saeed
Keyword(s):  
Temperature Field ◽  
Temperature Distribution ◽  
Hydration Heat ◽  
Maximum Temperature ◽  
Concrete Bridge ◽  
Central Temperature ◽  
Heat Temperature ◽  
Surface Cracking ◽  
Removal Time ◽  
Short Period

Measuring points were observed continuously to reveal the hydration heat temperature distribution of hollow concrete bridge pier. The results showed that as the thickness of the pier increased, the central temperature of the pier increased significantly due to hydration and the heat was difficult to be dissipated. The hydration temperature accounted for up to 70% of the maximum temperature rise during 20 h and reached the maximum temperature at 24 h after pouring the concrete. There was a jump value between the central temperature and surface temperature in a short period after removing the framework. The jumping was the most dangerous moment for the cracking of pier surfaces. Therefore, the formwork removal time has to be determined prudently and corresponding measures have to be conducted to reduce the possibility of pier surface cracking.

Theoretical Analysis of High Prandtl Number Heat Transfer in Non-Isothermal Pipe Flow

2005 ◽  
Author(s):  
Huajun Chen ◽  
Yitung Chen ◽  
Hsuan-Tsung Hsieh ◽  
Taide Tan
Keyword(s):  
Heat Transfer ◽  
Nusselt Number ◽  
Temperature Distribution ◽  
Analytical Solution ◽  
Prandtl Number ◽  
Pipe Flow ◽  
Fourier Expansion ◽  
Mathematical Expressions ◽  
High Prandtl Number ◽  
Turbulent Wall

Based on Fourier expansion, an analytical solution is developed for the high Prandtl number heat transfer in both fully developed laminar and turbulent non-isothermal pipe flow. Both of the mathematical expressions of the temperature distribution and the local Nusselt number have been obtained. A parametric study illustrates the characteristics of high Prandtl number heat transfer in non-isothermal pipe flow in detailed. The solutions obtained can be used for the numerical construction of the solution to the more general problems of heat transfer in the developed turbulent wall-bounded shear flows.

Heat transport into a shear flow at high Peclet number

1990 ◽  
Vol 427 (1872) ◽  
pp. 25-30
Keyword(s):  
Thermal Diffusivity ◽  
Temperature Distribution ◽  
Shear Flow ◽  
Heat Transport ◽  
Peclet Number ◽  
Asymptotic Form ◽  
Convex Region ◽  
Heated Region ◽  
Péclet Number ◽  
Special Case

Heat transport from a heated convex region on an otherwise insulating plane, into a fluid in shear flow along the plane, is considered. The asymptotic form of the temperature distribution is determined for large values of the Peclet number sL 2 / k where s is the shear rate of the flow, L is a typical dimension of the heated region and k is the thermal diffusivity of the fluid. From it the asymptotic form of the total heat transport is obtained. Although the shape of the region is arbitrary, the solution is constructed by using previous results for the special case of a heated strip with its edges normal to the flow.

Thermal Entrance Heat Transfer of an Adiabatically Prepared Fluid With Viscous Dissipation in a Tube With Isothermal Wall

2006 ◽  
Vol 128 (11) ◽  
pp. 1185-1193 ◽  
Author(s):  
A. Barletta ◽  
E. Magyari
Keyword(s):  
Temperature Distribution ◽  
Viscous Dissipation ◽  
Cross Section ◽  
Separation Of Variables ◽  
Upstream Region ◽  
Adiabatic Wall ◽  
Circular Duct ◽  
Isothermal Wall ◽  
Thermal Entrance ◽  
Special Case

Forced convection in the thermal entrance region of a circular duct is analyzed. Viscous dissipation effects are taken into account under conditions of laminar hydrodynamically developed flow. The duct wall is assumed to be isothermal in the region downstream of the entrance cross section. The prescription of the initial condition at the entrance cross section is coherent with the assumption of a non-negligible viscous heating in the whole duct. The special case of an adiabatic-wall preparation of the fluid in the upstream region is considered. This adiabatic preparation results in a non-uniform entrance temperature distribution. The governing equations are solved analytically by separation of variables. Important differences are pointed out in the comparison of the solution with those available in the literature, which are based on the assumption of a uniform temperature distribution in the entrance cross section.

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