Modelling temperature distribution

Tewkesbury, H., Stapley, A.G.F. and Fryer, P.J. (2000). Modelling temperature distributions in cooling chocolate moulds. Chemical Engineering Science, 55(16) 3123-3132. 

Tewkesbury, H., A.G.F. Stapley, and P.J. Fryer, Modelling Temperature Distribution in Cooling Chocolate Moulds, Chem. Engng. Sci. 55 3123-3132 (2000). 

Microscale 2D computation of an uncoated tool, 3 im edge radius. The solid model temperature distribution and output plots of x- and y-forces and peak tool temperature are evident. 

Takemasa, Y., S.Togati, and Y. Aral. 1996. Application of an unsteady-state model for predicting vertical temperature distribution to an existing atrium. ASHRAE Transactions, vol. 102, no. 1. 

In reality, heat is conducted in all three spatial dimensions. While specific building simulation codes can model the transient and steady-state two-dimensional temperature distribution in building structures using finite-difference or finite-elements methods, conduction is normally modeled one-... 

Most room models contain only one zone air node, thus assuming perfect mixing of the zone air and a homogenous temperature distribution in the space. Spatial temperature variations, such as vertical temperature gradients, are not considered. For specific applications such as displacement ventilation or atria, models with several zone air nodes in the vertical direction have been developed. ... 

Itiacd C., Ikmta H., Dalicieux P. Prediction of air temperature distribution in buildings with a zonal model. Energy and Buildings, vol. 24, 1996. 

II rime-dependent spatial airflow and temperature distributions in a room are requested, the coupling of CFD with a thermal model has to be considered. ... 

S. Boschert, P. Dold, K. W. Benz. Modelling of the temperature distribution in a three-zone resistance furnace influence of furnace configuration and ampoule position. J Cryst Growth 7S7 140, 1998. 

Fig. 21. Simplified model for temperature distribution in the combustion zone near a metallized solid propellant (D2).

From the start, we knew we needed large anodes to meet the challenge of inexpensive fluorine these calculations clearly show the need for a better design for large anodes. The obvious solution is to put a metal conductor down the middle of the anode. Figure 20 shows the results from a finite-element model of the temperature distribution in such an improved large anode with a central metal conductor. 

A variety of studies can be found in the literature for the solution of the convection heat transfer problem in micro-channels. Some of the analytical methods are very powerful, computationally very fast, and provide highly accurate results. Usually, their application is shown only for those channels and thermal boundary conditions for which solutions already exist, such as circular tube and parallel plates for constant heat flux or constant temperature thermal boundary conditions. The majority of experimental investigations are carried out under other thermal boundary conditions (e.g., experiments in rectangular and trapezoidal channels were conducted with heating only the bottom and/or the top of the channel). These experiments should be compared to solutions obtained for a given channel geometry at the same thermal boundary conditions. Results obtained in devices that are built up from a number of parallel micro-channels should account for heat flux and temperature distribution not only due to heat conduction in the streamwise direction but also conduction across the experimental set-up, and new computational models should be elaborated to compare the measurements with theory. 

The first approach developed by Hsu (1962) is widely used to determine ONE in conventional size channels and in micro-channels (Sato and Matsumura 1964 Davis and Anderson 1966 Celata et al. 1997 Qu and Mudawar 2002 Ghiaasiaan and Chedester 2002 Li and Cheng 2004 Liu et al. 2005). These models consider the behavior of a single bubble by solving the one-dimensional heat conduction equation with constant wall temperature as a boundary condition. The temperature distribution inside the surrounding liquid is the same as in the undisturbed near-wall flow, and the temperature of the embryo tip corresponds to the saturation temperature in the bubble 7s,b- The vapor temperature in the bubble can be determined from the Young-Laplace equation and the Clausius-Clapeyron equation (assuming a spherical bubble) ... 

Below we consider a quasi-one-dimensional model of flow and heat transfer in a heated capillary, with hydrodynamic, thermal and capillarity effects. We estimate the influence of heat transfer on steady-state laminar flow in a heated capillary, on the shape of the interface surface and the velocity and temperature distribution along the capillary axis. 

The present model takes into account how capillary, friction and gravity forces affect the flow development. The parameters which influence the flow mechanism are evaluated. In the frame of the quasi-one-dimensional model the theoretical description of the phenomena is based on the assumption of uniform parameter distribution over the cross-section of the liquid and vapor flows. With this approximation, the mass, thermal and momentum equations for the average parameters are used. These equations allow one to determine the velocity, pressure and temperature distributions along the capillary axis, the shape of the interface surface for various geometrical and regime parameters, as well as the influence of physical properties of the liquid and vapor, micro-channel size, initial temperature of the cooling liquid, wall heat flux and gravity on the flow and heat transfer characteristics. 

The quasi-one-dimensional model of flow in a heated micro-channel makes it possible to describe the fundamental features of two-phase capillary flow due to the heating and evaporation of the liquid. The approach developed allows one to estimate the effects of capillary, inertia, frictional and gravity forces on the shape of the interface surface, as well as the on velocity and temperature distributions. The results of the numerical solution of the system of one-dimensional mass, momentum, and energy conservation equations, and a detailed analysis of the hydrodynamic and thermal characteristic of the flow in heated capillary with evaporative interface surface have been carried out. 

A common feature in the models reviewed above was to calculate pressure and temperature distributions in a sequential procedure so that the interactions between temperature and other variables were ignored. It is therefore desirable to develop a numerical model that couples the solutions of pressure and temperature. The absence of such a model is mainly due to the excessive work required by the coupling computations and the difficulties in handling the numerical convergence problem. Wang et al. [27] combined the isothermal model proposed by Hu and Zhu [16,17] with the method proposed by Lai et al. for thermal analysis and presented a transient thermal mixed lubrication model. Pressure and temperature distributions are solved iteratively in a iterative loop so that the interactions between pressure and temperature can be examined. 

Effectively, Eqs. (86) and (87) describe two interpenetrating continua which are thermally coupled. The value of the heat transfer coefficient a depends on the specific shape of the channels considered suitable correlations have been determined for circular or for rectangular channels [100]. In general, the temperature fields obtained from Eqs. (86) and (87) for the solid and the fluid phases are different, in contrast to the assumptions made in most other models for heat transfer in porous media [117]. Kim et al. [118] have used a model similar to that described here to compute the temperature distribution in a micro channel heat sink. They considered various values of the channel width (expressed in dimensionless form as the Darcy number) and various ratios of the solid and fluid thermal conductivity and determined the regimes where major deviations of the fluid temperature from the solid temperature are found. 

GP 2] [R 2] The radial temperature distribution was determined by modeling, using a worst-case scenario (5 Nl h stoichiometric mixture without inert 100% conversion 80% selectivity) [102], The maximum radial temperature difference amounts to approximately 0.5 K. Thus, isothermal behavior in the radial direction can be diagnosed. 

Non-uniform temperature distribution in a reactor assumed model based on the Fourier heat conduction in an isotropic medium equality of temperatures of the medium and the surroundings assumed at the boundary critical values of Frank-Kamenetskii number given. 

In some cases, it may be of interest to model the temperature distribution through the wall. 

Figure 3.9. Model representation of temperature distribution in the wall and jacket, showing wall and Jacket with four lumped parameters.

The spatial temperature distribution established under steady-state conditions is the result both of thermal conduction in the fluid and in the matrix material and of convective flow. Figure 2. 9.10, top row, shows temperature maps representing this combined effect in a random-site percolation cluster. The convection rolls distorted by the flow obstacles in the model object are represented by the velocity maps in Figure 2.9.10. All experimental data (left column) were recorded with the NMR methods described above, and compare well with the simulated data obtained with the aid of the FLUENT 5.5.1 [40] software package (right-hand column). Details both of the experimental set-up and the numerical simulations can be found in Ref. [8], The spatial resolution is limited by the same restrictions associated with spin... 

A simplified version of the model in Table IX, neglecting accumulation of mass and heat as well as dispersion and conduction in the gas phase, predicts dynamic performance of a laboratory S02 converter operating under periodic reversal of flow direction quite well. This is shown by Fig. 13 taken from Wu et al. (1996). Data show the temperature profiles in a 2-m bed of the Chinese S101 catalyst once a stationary cycling state is attained. One set of curves shows the temperature distribution just after switching direction and the second shows the distribution after a further 60 min. Simulated and experimental profiles are close. The surprising result is that the experimental maximum temperatures equal or exceed the simu-... 

Several features of the early model (Fig. 6) have been modified in the present-day, high-temperature version of this calorimeter (Fig. 7) (37). Depending upon the temperature range envisaged, the block is made of refractory steel, alumina, or beryllium oxide and is machined to house the calorimeter itself. The thermoelectric pile (about 50 platinum to platinum-rhodium thermocouples) is affixed in the grooves of an alumina plate (A), which is permanently cemented to two cylindrical tubes of alumina (B). Cylindrical containers of platinum (C) ensure the uniformity of the temperature distribution within the calorimeter cells. 

The heat-transfer model described in Sections IV.A.3 and IV.B.2 can be applied to calculate the temperature distribution inside the droplet and energy... 

Two classical models have been described for runaway calculations in which the important difference between the two is in the degree of mixing. The first model, proposed by Semenov [165], applies to well stirred mixtures where the temperature is the same throughout the mixture. Heat removal occurs with a steep temperature gradient at the surface of the walls or coils, and is governed by the usual factors of area, temperature of coolant, and heat transfer coefficients. Case A in Figure 3.20 shows a temperature distribution by the Semenov model for self-heating. 

The second model, proposed by Frank-Kamenetskii [162], applies to cases of solids and unstirred liquids. This model is often used for liquids in storage. Here, it is assumed that heat is lost by conduction through the material to tire walls (at ambient temperature) where the heat loss is infinite compared to the rate of heat conduction through the material. The thermal conductivity of the material is an important factor for calculations using this model. Shape is also important in this model and different factors are used for slabs, spheres, and cylinders. Case B in Figure 3.20 indicates a typical temperature distribution by the Frank-Kamenetskii model, showing a temperature maximum in the center of the material. 

The more recent Thomas model [209] comprises elements of both the Semenov and Frank-Kamenetskii models in that there is a nonuniform temperature distribution in the liquid and a steep temperature gradient at the wall. Case C in Figure 3.20 shows a temperature distribution curve from self-heating for the Thomas model. The appropriate model (Semenov, Frank-Kamenetskii, or Thomas) is determined by the ratio of the heat removal from the vessel and the thermal conductivity in the vessel. This ratio is determined by the Biot number (Nm) which has been described previously as hx/X, in which h is the film heat transfer coefficient to the surroundings (air, cooling mantle, etc.), x is the distance such as the radius of the vessel, and X is the effective thermal conductivity. 

FIGURE 3.20. Typical Temperature Distributions during Self-Heating in a Vessel. A = The Semenov Model B = The Frank-Kamenetskii Model C = The Thomas Model... 

Nevertheless, these modeling efforts are of little value when the practical implementation does not corroborate the above-calculated results. Ensuring the constancy of any parameter in the catalytic testing workflow, the reactor performance with regard to temperature distribution, gas distribution, constant feed... 

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