Steady-state heat transfer

The overall heat transfer coefficient, U, is a measure of the conductivity of all the materials between the hot and cold streams. For steady state heat transfer through the convective film on the outside of the exchanger pipe, across the pipe wall and through the convective film on the inside of the convective pipe, the overall heat transfer coefficient may be stated as ... 

Show that the steady state heat transfer rate is given by the following equation. 

The two steady-state heat-transfer coefficients, hr and hj, could be further described in terms of the physical properties of the system. The solution-to-wall coefficient for heat transfer, hT in Equation 8.8, is strongly dependent on the physical properties of the reaction mixture (heat capacity, density, viscosity and thermal conductivity) as well as on the fluid dynamics inside the reactor. Similarly, the wall-to-jacket coefficient for heat transfer, hj, depends on the properties and on the fluid dynamics of the chosen cooling liquid. Thus, U generally varies during measurements on a chemical reaction mainly for the following two reasons. 

The variability of physical properties widens both the dimensional x- and the dimensionless pi-space. The process is not determined by the original material quantity x, but by its dimensionless reproduction. (Pawlowski [27] has clearly demonstrated this situation by the mathematical formulation of the steady-state heat transfer in an concentric cylinder viscometer exhibiting Couette flow). It is therefore important to carry out the dimensional-analytical reproduction of the material function uniformly in order to discover possibly existing, but under circumstances concealed, similarity in the behavior of different substances. This can be achieved only by the standard representation of the material function [5, 27]. 

We will start by discussing a pi-space in which an optional process in an apparatus of a given geometry takes place and where the hydrodynamics is coupled with a steady-state heat transfer [35]. The target pi-number will be laid down later on (Example 17). In case temperature independent materials take part, the process will be described by the following possible process-related and material pi-numbers ... 

In his famous and extremely short essay entitled The principle of similitude Lord Rayleigh [4] discussed 15 different physical problems which can be condensed to laws by using dimensional analysis without performing any experiments. The last of these examples concerned the Boussinesq problem of the steady-state heat transfer from a fixed body to a ideal liquid flowing with the velocity of v. 

Example 35 Steady-state heat transfer in bubble columns... 

Example 35 Steady-state heat transfer in bubble columns 149 Example 36 Time course of temperature equalization in a liquid with temperature-dependent viscosity in the case of free convection 153 Example 37 Mass transfer in stirring vessels in the G/L system (bulk aeration) Effects of coalescence behavior of the material system 156 Example 38 Mass transfer in the G/L system in bubble columns with injectors as gas distributors. The effects of coalescence behavior of the material system 160... 

Steady state heat transfer refers to the condition where the rate of heat flowing into one face of an object is equal to that flowing out of the other. If, for example, a slab of metal were placed on a hot-plate, the heat flowing into the metal would initially contribute to a temperature rise in the material, until ultimately a linear temperature gradient formed between the hot and cold faces, wherein heat flowing in would equal heat flowing out and steady state heat transfer would be established. The time involved before steady state conditions axe encountered is dependent on the thermal requirements, that is, the total heat capacity of the material. A useful constant, therefore, in depicting transient, or non-steady state heat transfer is the thermal diffusivity ... 

Fourier s law for steady state heat transfer can be translated to the cylindrical geometry sketched in Figure 9.1. Recognizing that the surface area of a cylinder is 2nrL ... 

Consider the one-dimensional, steady-state heat transfer for composite structure consisting of parallel plates, coaxial cylinders, etc., in perfect thermal contact with each other. 

Consider steady-state heat transfer in a rectangular cross section, with constant thermal conductivity. Design the four nonhomogeneous boundary conditions such that the final expression of the temperature distribution consists only of four terms and not an infinite series of terms. Write down this expression. (Hint Look at the solutions of Ex. 4.3 and Prob. 4.9.)... 

Thermal conductivity is the intensive property of a material that indicates its ability to conduct heat. For one-dimensional heat flow in the x-direction the steady state heat transfer can be described by Fourier s law of heat conduction ... 

In Chap. 2 steady-state heat transfer was calculated in systems in which the temperature gradient and area could be expressed in terms of one space coordinate. We now wish to analyze the more general case of two-dimensional heat flow. For steady state, the Laplace equation applies. 

Figure 3.46 presents the temperature distribution in the plane y = 0, which separates the left parts from the right parts of the bricks assembly. The shape of the group of the isothermal curves shows a displacement towards the brick with the higher thermal conductivity. Using the values obtained from these isothermal curves, it is not difficult to establish that the exit heat flux for each brick from the bottom of the assembly (plane Z = — 1 ) and for the top of the assembly (plane Z = 1) depends on its thermal conductivity and on the distribution of the isothermal curves. If we compare this figure to Fig. 3.47 we can observe that the data contained in Fig. 3.46 correspond to the situation of a steady state heat transfer. 

Now let us envision a power level so low that the steady-state heat transfer occurs at some temperature just below and no matter how long we apply power, we transfer it away just fast enough that T cannot rise any further. In this case, we could not initiate the explosive. We will call the power at this condition Pq. Since we are at steady-state heat transfer, the temperature is constant and dT/dt in Eq. (23.2) must equal zero therefore Pq = XT. 

In order to size the compressor of a new refrigerator, it is desired to determine the rate of heat transfer from the kitchen air into the refrigerated space through the walls, door, and the top and bottom section of the refrigerator (Fig. 2-11). In your analysis, would you treat this as a transient or steady-state heat transfer problem Also, would you consider the heat transfer to be one-dimensional or multidimensional Explain. 

In these expressions, p denotes the density of cordicnte ceramic (41.15 g/in. ), P is the fractional porosity of the cell wall, and Cp is the specific heat of the cell wall (0 25 cal/ g°C). TIF is a measure of the temperature gradient the substrate can withstand prior to fracture MIF is a measure of the crush strength of the substrate in the diagonal direction Rf IS a measure of back pressure, H, is a measure of steady-state heat transfer, and LOF IS a measure of light-off performance An ideal substrate must offer high GSA, OFA, TIF, MIF, H and LOF values and low D, p, and Rf values. A close examination of the expressions indicates that certain compromises are necessary in arriving at the optimum substrate, as discussed in the next section. 

Steady State Heat Transfer in Micro Conduits... 

Consider steady-state heat transfer in thermally developing, hydrodynamically developed forced laminar flow inside a micro conduits (parallel plate micro channel or micro mbe) under following assumptions o The fluid is incompressible with constant physical properties, o The free heat convection is negligible, o The energy generation is negligible, o The entrance temperature is uniform, o The surface temperature is uniform. 

Conversely, the heat flow promoted by a given temperature difference is reduced if the thermal resistance is increased. This is the principle of insulation by lagging, and it is illustrated by a composite wall, as shown in Fig. 4B. If steady-state heat transfer exists, the rate of heat transfer is the same for both materials. Therefore,... 

ASTM C 335-89 Standard Test Method for Steady-State Heat Transfer Properties of Horizontal Pipe Insulation, 10 pp (Comm C-16)... 

In most steady-state heat transfer problems, more than one heat transfer mode may be involved. The various thermal resistances due to thermal convection or conduction may be combined and described by an overall heat transfer coefficient, U. Using U, the heat transfer rate, Q, can be calculated from the terminal and/or system temperatures. The analysis of this problem is simplified when the concepts of thermal circuit and thermal resistance are employed. 

Consider the steady state heat transfer problem.[l] The boundary condition at y = 1 (the nonhomogeneous boundary condition) is used to find the coefficient in the infinite series after separating the variables. 

Consider the steady state heat transfer problem with nonhomogeneous boundary conditions in both x and y... 

Solve the following steady state heat transfer problem by applying the Laplace transformation in y coordinate ... 

Consider a nonlinear steady state heat transfer problem governed by the following elliptic PDE ... 

For steady-state heat transfer within a planar film, the energy balance relation (Eq. 11.1.1) simplifies to... 

I. Steady-State Heat Transfer in Fully Developed Flow through a Heated (or Cooled) Section of a Circular Tube... 

The removal of precipitated polyethylene from the wall is an interesting operation. About once every 2-3 sec the expansion valve is opened more fully than required for the expansion/precipitation function this results in a rapid decrease in pressure in the reactor of as much as 300-600 bar. The concomitant rapid increase in the velocity of the gas phase in the tubular reactor shears the walls and strips off any deposited polyethylene so that a reasonably steady state heat transfer situation exists. This description of the operation of the polymerization process, the polyethylene precipitation step, and the accentuated expansion, which maintains a clean wall and a high heat transfer coefficient, help to illustrate the interesting SCF solubility behavior and they also supply some information on the commercial reality of high-pressure processing in what we consider to be an extreme case. 

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