Heat transfer equation solutions

At a later stage of bubble growth, heat diffusion effects are controlling (as point c in Fig. 2.9), and the solution to the coupled momentum and heat transfer equations leads to the asymptotic solutions and is closely approximated by the leading term of the Plesset-Zwick (1954) solution,... 

The solution of the heat transfer equation with f = F0 (constant) at y = 0 when t > 0... 

Solution of condensed-phase heat transfer equation is needed to analyze structural response to fires and simulate flame spread on solid surfaces. The solution of this conjugate heat transfer problem simulate is typical for fires, but rarely found in commercial CFD packages. Over the years, different techniques have been developed to tackle this problem. Since solid-phase heat transfer... 

This is Fick s second diffusion equation [242], an adaptation to diffusion of the heat transfer equation of Fourier [253]. Technically, it is a second-order parabolic partial differential equation (pde). In fact, it will mostly be only the skeleton of the actual equation one needs to solve there will usually be such complications as convection (solution moving) and chemical reactions taking place in the solution, which will cause concentration changes in addition to diffusion itself. Numerical solution may then be the only way we can get numbers from such equations - hence digital simulation. 

C vs. the start at 100°C. Thus we have a significant effect of the level of activity upon the magnitude of intraparticle temperatures, but not upon the time response. None of this behavior shows up in the Prater Number. The key to understanding time response comes fi om a detailed solution of the transient heat transfer equation. The dimensionless time will take the form (Lee et al. [16]) ... 

In laminar flow, heat transfer occurs only by conduction, as there are no eddies to carry heat by convection across an isothermal surface. The problem is amenable to mathematical analysis based on the partial differential equations for continuity, momentum, and energy. Such treatments are beyond the scope of this book and are given in standard treatises on heat transfer, Mathematical solutions depend on the boundary conditions established to define the conditions of fluid flow and heat transfer. When the fluid approaches the heating surface, it may have an already completed hydrodynamic boundary layer or a partially developed one. Or the fluid may approach the heating surface at a uniform velocity, and both boimdary layers may be initiated at the same time. A simple flow situation where the velocity is assumed constant in all cross sections and tube lengths is called... 

The temperature distribution in the lithosphere (Figs. 6.2,6.3) is obtained by solution of the heat transfer equation derived in the frame of the energy balance scheme (Carlslaw and Jaeger 1959 Paskonov et al. 1988) ... 

The symbols used in this formula are the same as in the previous section, the vibrational temperature of molecular gas is assumed to be high > hco). Solution of the heat transfer equation with the flux (7-13) from the surface (x = 0) taken as a boundary condition is (Bochin, Rusanov, Lukianchuk, 1976),... 

Procedure of Obtaining the Solutions to Heat Transfer Equations Representing the Dynamics of Temperature Change In a Rolling Tire... 

Iterative Solution of the Heat Transfer Equation by Finite Difference Approximations... 

The heating of a viscous fluid in laminar flow in a tube of radius R (diameter, D) will now be considered. Prior to the entry plane z < 0), the fluid temperature is uniform at Tf for z > 0, the temperature of the fluid will vary in both radial and axial directions as a result of heat transfer at the tube wall. A thermal energy balance will first be made on a differential fluid element to derive the basic governing equation for heat transfer. The solution of this equation for the power-law and the Bingham plastic models will then be presented. 

The calculation method is that described in Jakob (1962) which uses the finite difference method for the solution of the heat transfer equations. The concrete layers have been grouped in a certain number of groups, each with an average thickness and an exposed surface equal to the sum of the surfaces of the concrete layers included in the group. 

The reactor can be assumed to be a two-dimensional axi-symmetric vessel so any vertical section taken through the vessel will have the same flow pattern [5-9]. Strictly, this will only apply if there are no internal mechanical components. If baffles or internal cooling coils are used it is necessary to use a full three-dimensional model [10-12]. Unfortunately, this adds considerably to the computational effort to obtain a solution. For the case considered here a computational technique based on the use of finite elements is used to solve the flow and heat transfer equations so that a wide variety of vessel geometries can be examined. 

Laminar Flow Although heat-transfer coefficients for laminar flow are considerably smaller than for turbulent flow, it is sometimes necessary to accept lower heat transfer in order to reduce pumping costs. The heat-flow mechanism in purely laminar flow is conduction. The rate of heat flow between the walls of the conduit and the fluid flowing in it can be obtained analytically. But to obtain a solution it is necessary to know or assume the velocity distribution in the conduit. In fully developed laminar flow without heat transfer, the velocity distribution at any cross section has the shape of a parabola. The velocity profile in laminar flow usually becomes fully established much more rapidly than the temperature profile. Heat-transfer equations based on the assumption of a parabolic velocity distribution will therefore not introduce serious errors for viscous fluids flowing in long ducts, if they are modified to account for effects caused by the variation of the viscosity due to the temperature gradient. The equation below can be used to predict heat transfer in laminar flow. 

Figure 8.1 presupposes that the kiln radius is large compared to the penetration depth, d(0), that develops during contact of the heated wall coming from the exposed wall with the bed. We can set the initial conditions and boundary conditions for the solution of a two-dimensional transient heat transfer equation, similar to Equation (8.5), using the time variable in terms of the rotation as f = (0 + g/2)/2ir( ), which is zero at the instant of initial wall-to-bed contact. The final time is t = /2ttco when the wall exits the covered bed. Between the time of initial contact and the maximum time, y = R-r where R is the kiln radius and r is the radius at the penetration depth, that is. 

Based on the effective kinetics and heat transfer parameters, solution of the differential equations by numerical integration in the radial and axial direction is carried out by programs, for example such as Presto-Kinetws (CiT, http //www.cit-wulkow.de/), which was frequently used in this book. 

The solution of the neutronic equations yields the power generation distribution and the neutron multiplication factor. The solution of the heat transfer equations yields the corresponding temperature... 

The solution of the heat transfer equations is largely controlled by the boundary conditions used to specify the problem. In solidification problems, these boundary conditions depend on the nature of the contact between the freezing material and its container as well as the heat transferred by the container to the external cooling media. The latter problem has been well documented in the heat transfer literature over the years where correlations for convective heat transfer to flowing streams of coolant, natural convection and so on abound. The boundary conditions between the freezing material and mold however, are less well documented. 

Solution. The temperature variation in the polymeric melt is described by the one-dimensional heat transfer equation... 

In the following discussion, attention is focused on the application of thermal-hydraulic system codes. Under this category codes like APROS, ATHLET, CATHARE, RELAP5 and TRAC are included, all based upon the solution of a main system of six partial differential equations. Two main fields, one per each of the two phases liquid and steam are considered and coupling is available with the solution of the conduction heat transfer equations within solids interfaced with the fluid phases. A one-dimensional solution for the characteristics of the fluid is achieved in the direction of the fluid motion in time dependent conditions. It should be emphasized that more sophisticate models are also available including three-dimensional solutions and multi-field approaches in two and multiphase fluids. However the present qualification level of those sophisticated computational tools is questionable as well as their actual need in the design or in the safety applications. 

In the finite element solution of the energy equation it is sometimes necessary to impose heat transfer across a section of the domain wall as a boundary condition in the process model. This type of convection (Robins) boundary condition is given as... 

The definition of the heat-transfer coefficient is arbitrary, depending on whether bulk-fluid temperature, centerline temperature, or some other reference temperature is used for ti or t-. Equation (5-24) is an expression of Newtons law of cooling and incorporates all the complexities involved in the solution of Eq. (5-23). The temperature gradients in both the fluid and the adjacent solid at the fluid-solid interface may also be related to the heat-transfer coefficient ... 

Natural convection occurs when a solid surface is in contact with a fluid of different temperature from the surface. Density differences provide the body force required to move the flmd. Theoretical analyses of natural convection require the simultaneous solution of the coupled equations of motion and energy. Details of theoretical studies are available in several general references (Brown and Marco, Introduction to Heat Transfer, 3d ed., McGraw-HiU, New York, 1958 and Jakob, Heat Transfer, Wiley, New York, vol. 1, 1949 vol. 2, 1957) but have generally been applied successfully to the simple case of a vertical plate. Solution of the motion and energy equations gives temperature and velocity fields from which heat-transfer coefficients may be derived. The general type of equation obtained is the so-called Nusselt equation hL I L p gp At cjl... 

Analogy between Momentum and Heat Transfer The interrelationship of momentum transfer and heat transfer is obvious from examining the equations of motion and energy. For constant flmd properties, the equations of motion must be solved before the energy equation is solved. If flmd properties are not constant, the equations are coupled, and their solutions must proceed simultaneously. Con-... 

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