Heat transfer exact solutions

As we have seen before, exact differentials correspond to the total differential of a state function, while inexact differentials are associated with quantities that are not state functions, but are path-dependent. Caratheodory proved a purely mathematical theorem, with no reference to physical systems, that establishes the condition for the existence of an integrating denominator for differential expressions of the form of equation (2.44). Called the Caratheodory theorem, it asserts that an integrating denominator exists for Pfaffian differentials, Sq, when there exist final states specified by ( V, ... x )j that are inaccessible from some initial state (.vj,.... v )in by a path for which Sq = 0. Such paths are called solution curves of the differential expression The connection from the purely mathematical realm to thermodynamic systems is established by recognizing that we can express the differential expressions for heat transfer during a reversible thermodynamic process, 6qrey as Pfaffian differentials of the form given by equation (2.44). Then, solution curves (for which Sqrev = 0) correspond to reversible adiabatic processes in which no heat is absorbed or released. 

To identify the governing processing and material parameters, a one dimensional case was analyzed. The heat transfer problem renders an exact solution, [10], which can be presented as an infinite series... 

Exact analytical solutions of the coupled equations for simultaneous mass transfer, heat transfer, and chemical reaction cannot be obtained. However, various authors have employed linear approximations (56-57), perturbation techniques (58), or asymptotic approaches (59) to obtain approximate analytical solutions to these equations. Numerical solutions have also been obtained (60-61). Once the solution for the concentration profile has been determined, equation 12.3.98 may be used to determine the temperature profile. The effectiveness factor may also be determined from the concentration profile, using the approach we have... 

We follow the analysis of Frank-Kamenetskii [3] of a slab of half-thickness, rG, heated by convection with a constant convective heat transfer coefficient, h, from an ambient of Too. The initial temperature is 7j < 7 ,XJ however, we consider no solution over time. We only examine the steady state solution, and look for conditions where it is not valid. If we return to the analysis for autoignition, under a uniform temperature state (see the Semenov model in Section 4.3) we saw that a critical state exists that was just on the fringe of valid steady solutions. Physically, this means that as the self-heating proceeds, there is a state of relatively low temperature where a steady condition is sustained. This is like the warm bag of mulch where the interior is a slightly higher temperature than the ambient. The exothermiscity is exactly balanced by the heat conducted away from the interior. However, under some critical condition of size (rG) or ambient heating (h and Too), we might leave the content world of steady state and a dynamic condition will... 

In what follows, the preceding evaluation procedure is employed in a somewhat different mode, the main objective now being to obtain expressions for the heat or mass transfer coefficient in complex situations on the basis of information available for some simpler asymptotic cases. The order-of-magnitude procedure replaces the convective diffusion equation by an algebraic equation whose coefficients are determined from exact solutions available in simpler limiting cases [13,14]. Various cases involving free convection, forced convection, mixed convection, diffusion with reaction, convective diffusion with reaction, turbulent mass transfer with chemical reaction, and unsteady heat transfer are examined to demonstrate the usefulness of this simple approach. There are, of course, cases, such as the one treated earlier, in which the constants cannot be obtained because exact solutions are not available even for simpler limiting cases. In such cases, the procedure is still useful to correlate experimental data if the constants are determined on the basis of those data. 

This is a functional equation for the boundary position X and the unknown constant parameter n. Upon substituting Eq. (256) into Eq. (251) an ordinary differential equation is obtained for X(t, n), and a family of curves in the phase plane (X, X) can be obtained. For n sufficiently close to unity two functions in the phase plane can be determined which serve as upper and lower bounds for the trajectories. The choice is guided by reference to the exact solution for the limiting case of constant surface temperature. It is shown that the upper and lower bounds are quite close to the one-parameter phase plane solution, although no comparison is made with a direct numerical solution. The one-parameter solution also agrees well with experiments on the solidification of aluminum under conditions of low surface heat transfer coefficient (hi = 0.02 cm.-1). 

As indicated in Table 7.10, only in the last decade have models considered all three phenomena of heat transfer, fluid flow, and hydrate dissociation kinetics. The rightmost column in Table 7.10 indicates whether the model has an exact solution (analytical) or an approximate (numerical) solution. Analytic models can be used to show the mechanisms for dissociation. For example, a thorough analytical study (Hong and Pooladi-Danish, 2005) suggested that (1) convective heat transfer was not important, (2) in order for kinetics to be important, the kinetic rate constant would have to be reduced by more than 2-3 orders of magnitude, and (3) fluid flow will almost never control hydrate dissociation rates. Instead conductive heat flow controls hydrate dissociation. 

When a liquid warms up, its density decreases, which results in buoyancy and an ascendant flow is induced. Thus, a reactive liquid will flow upwards in the center of a container and flow downwards at the walls, where it cools this flow is called natural convection. Thus, at the wall, heat exchange may occur to a certain degree. This situation may correspond to a stirred tank reactor after loss of agitation. The exact mathematical description requires the simultaneous solution of heat and impulse transfer equations. Nevertheless, it is possible to use a simplified approach based on physical similitude. The mode of heat transfer within a fluid can be characterized by a dimensionless criterion, the Rayleigh number (Ra). As the Reynolds number does for forced convection, the Rayleigh number characterizes the flow regime in natural convection ... 

Figure 3 indicates that in the example, the trade-off curve (or the non-inferior solutions) is not completely concave in shape and that the feasible region is not exactly convex however, the two objectives, capital investment, in terms of the heat transfer area and available energy, f, are always in conflict with each other in the region under consideration. A reduction in the heat transfer area will always give rise to an increase in the dissipation of available energy. 

Comparison of the results of the integral method with experimental results and exact solutions show that the prediction of the heat transfer coefficient with the integral method is satisfactory. 

A R Shouman. An Exact General Solution for the Temperature Distribution and the Radiation Heat Transfer Along a Constant Cross-Sectional Area Fin. Paper No. 67-WA/HT-27, ASME, Nov 1967. 

The solution to the two equations will have exactly the same form when a = v. Thus we should expect that the relative magnitudes of the thermal diffusivity and kinematic viscosity would have an important influence on convection heat transfer since these magnitudes relate the velocity distribution to the temperature distribution. This is exactly the case, and we shall see the role which these parameters play in the subsequent discussion. 

Transfer exactly one-tenth of the total net weight of the dried sample (representing 0.5 g of the original, unwashed sample) to a 50-mL beaker, and wet with about 2 mL of ethanol. Dissolve the sample in 25 mL of 0.125 M sodium hydroxide. Agitate the solution for 1 h at room temperature. Quantitatively transfer the saponified sample solution to a 50-mL volumetric flask, and dilute to volume with water. Transfer 25.0 mL of the diluted sample solution to the round-bottom flask of the distillation apparatus, and add 20 mL of Clark s solution. Start the distillation by heating the round-bottom flask. Collect the first 15 mL of distillate separately in a measuring cylinder. Then start the steam supply, and continue distillation until 150 mL of distillate has been collected in a 200-mL beaker. Quantitatively combine the distillates, titrate with 0.05 M sodium hydroxide to pH 8.5, and record the volume of titrant required, in milliliters, as S. 

The major problem in temperature control in bulk and solution batch chain-growth reactions is the large increase in viscosity of the reaction medium with conversion. The viscosity of styrene mixtures at I50°C will have increased about 1000-fold, for example, when 40 wt % of the monomer has polymerized. The heat transfer to a jacket in a vessel varies approximately inversely with the one-third power of the viscosity. (The exact dependence depends also on the nature of the agitator and the speed of fluid flow.) This suggests that the heat transfer efficiency in a jacketed batch reactor can be expected to decrease by about 40% for every 10% increase in polystyrene conversion between 0 and 40%. 

Boundary-layer theory has been applied to solve the heat-transfer problem in forced convection laminar flow along a heated plate. The method is described in detail in numerous textbooks (El, G5, S3). Some exact solutions and approximate solutions are also obtained (B2, S3). 

This solution of the FK equation represents the condition where the rate of heat evolved exactly equals the rate at which it is transferred away for a given size and shape of an explosive charge. Any increase in ambient temperature above Tq would lead to a runaway reaction or explosion within a finite time. 

This is certainly a very desirable form of solution since the temperature at any point within the sphere can be determined simply by substituting the r-coordinate of the point into the analytical solution function above. The analytical solution of a problem is also referred to as the exact solution since it satisfies the differential equation and the boundary conditions. This can be verified by substituting the solution function into the differential equation and the boundary conditions. Further, the rate of heat transfer at any location within the sphere or its surface can be determined by taking the derivative of the solution function T r) and substituting it into Fourier s law as... 

Consider the variation of the solution of a transient heat transfer problem with time at a specified nodal point. Both the numerical and actual (exact) solutions coincide at the beginning of the first time step, as expected, but the numerical. solution deviates from the exact. solution as the time t increases. The difference between the two solutions at t Ar is due to the approximation at the first time step only and is called the local discretization error. One would expect the situation to get worse with each step since the second step uses the erroneous result of the first step a,s its starting point and adds a second local discretization error on top of it, as shown... 

At Reynolds numbers greater than about 30, it is observed that waves form at the liquid-vapor interface although the flow in liquid film remains laminar. I he flow in this case is said to be wavy laminar. The waves at the liquid-vapor interface tend to increase heat transfer. But the waves also complicate the analysis and make it very difficult to obtain analytical solutions. Therefore, we have to rely on experimental studies. The increase in heat transfer due to the wave effect is, on average, about 20 percent, but it can exceed 50 percent. The exact amount of enhancement depends on the Reynolds number. Rased on his experimental studies, Kutateladze (1963) recommended the following relation for the average heat transfer coefficient in wavy laminar condensate flow for p p, and 30 < Re < 1800,... 

Kakag S., Y. Yener, 1973, Exact solution of the transient forced convection energy equation for time wise variation of inlet temperature, Int. J. Heat Mass Transfer 16, 2205-2214. 

For k —> oo the temperature at x = 0 has the value i) i) ]. This is the boundary condition in the Stefan problem discussed in 2.3.6.1. From (2.218) we obtain the first term of the exact solution (2.213), corresponding to Ph — oo, and therefore the time t according to (2.214). With finite heat transfer resistance (1 /k) the solidification-time is greater than i it no longer increases proportionally to s2. 

As a comparison with the exact solution of the Stefan problem shows, the quasisteady approximation discussed in the last section only holds for sufficiently large values of the phase transition number, around Ph > 7. There are no exact solutions for solidification problems with finite overall heat transfer resistances to the cooling liquid or for problems involving cylindrical or spherical geometry, and therefore we have to rely on the quasi-steady approximation. An improvement to this approach in which the heat stored in the solidified layer is at least approximately considered, is desired and was given in different investigations. 

Heat transfer of packed bed has been the subject of numerous studies. For cylindrical packed columns, a solution for determining temperature distributions was given using Bessel functions. Here, it is important to find out exact effective thermal conductivity of bed because of flowing gas and relatively high temperatures. Radial temperature distributions are more important than that of axial direction because the latter can be measured and controlled during the operation. 

We then return briefly to consider the creeping-flow approximation of the previous two chapters. We do this at this point because we recognize that the creeping-flow solution is exactly analogous to the pure conduction heat transfer solution of the preceding section and thus should also not be a uniformly valid first approximation to flow at low Reynolds number. We thus explain the sense in which the creeping-flow solution can be accepted as a first approximation (i.e., why does it play the important role in the analysis of viscous flows that it does ), and in principle how it might be corrected to account for convection of momentum (or vorticity) for the realistic case of flows in which Re is small but nonzero. 

Even with these simplifications, however, it is rarely possible to obtain analytic solutions for fluid mechanics or heat transfer problems. The Navier Stokes equation for an isothermal fluid is still nonlinear, as can be seen by examination of either (2 89) or (2 91). The Bousi-nesq equations involve a coupling between u and 6, introducing additional nonlinearities. It will be noted, however, that, provided the density can be taken as constant in the body-force term (thus neglecting any natural convection), the fluid mechanics problem is decoupled from the thermal problem in the sense that the equations of motion, (2 89) or (2-91), and continuity, (2-20), do not involve the temperature 0. The thermal energy equation, (2-93), is actually a linear equation in the unknown 6, once the Boussinesq approximation has been introduced. In that case, the only nonlinear term is dissipation, but this involves the product E E and can be treated simply as a source term that will be known once Eqs. (2-89) or (2 91) and (2 20) have been solved to determine the velocity. In spite of being linear, however, the velocity u appears as a coefficient (in the convective derivative term). Even when the form of u is known (either exactly or approximately), it is normally quite a complicated function, and this makes it extremely difficult to obtain analytic solutions for 0 even though the governing equation is linear. 

To evaluate heat transfer rates by means of the definition (9-12) or (9-10), it is necessary to solve (9-7) and (9-8). Although this problem is linear in 6, the coefficient u in (9-7) is almost always too complex to allow exact analytic solutions for 6. The best that we can do, analytically, is to solve (9-7) approximately for limiting values of Re and Pe. 

The forced convection heat transfer problem [Eq. (9-7) plus boundary conditions] is linear in 6, but it still cannot be solved exactly (except for special cases) for Pe > 0(1) because of the complexity of the coefficient u. What may appear surprising at first is that simplifications arise in the limit Pe 1, which allow an approximate solution even though no analytic solutions (exact or approximate) are possible for intermediate values of Pe. This is surprising because the importance of the troublesome convection term, which is... 

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