Rating equations, heat exchanger

The objective of a rating problem is to determine if an existing process unit will satisfy process conditions. To arrive at an approximate calculation procedure for rating a heat exchanger, first define a clean overall heat-transfer coefficient, i.e., in the absence of any fouling. Therefore, Rfj andRfo = 0 in Equation 4.15. 

Table 4.7 Summary of Equations for Rating a Heat Exchanger ...

In this section we shall be concerned with more realistic models of tubular reactors. The isothermal reactor is obviously the simplest type, but it implies that either there are no large heat effects or that they can be completely dominated by temperature control. The reactor with an optimal temperature profile is clearly the most desirable, but this means that the rate of heat exchange can be regulated precisely at each point. Between these two extremes there is a range of designs about which something should be said. We shall not always solve the equations in detail but we shall try to show the important features of the behavior of the reactor by means of examples. 

Below we develop a simple model of heat transport in a stack element shown in Figure 5.2 (Kulikovsky, 2009d). An analysis of dimensionless equations reveals the dominant term the rate of heat exchange between the BP and air in the channel. This allows us to construct an asymptotic solution to a problem. 

The divergence of equals the rate of heat exchange with the BP. The equation of heat balance in the channel, therefore, is... 

In view of Eq. (5.26), 1/e thanks to the smallness of e, parameter is large ( > 10, Table 5.2) and 10. Thus, the leading term in the heat transport equations is the rate of heat exchange between air in the channel and the stack. Due to this high rate, T and Tair are close to each other and their -dependencies are well described by a simple linear law. A numerical model (lora et al., 2005) supports this conclusion. 

Equation (5.72) does not include the interconnect thermal conductivity. Physically, this parameter plays a minor role in the heat balance due to the dominating rate of heat exchange between the interconnect and air flow. Note also that the temperature gradient over the channel length increases with the growth of inlet air flow temperature. ... 

Dividing Eq. (5.107) by and equating 7 to zero we see that all terms except the last one vanish and a zero-order solution to (5.107) is T = Tj. Thus, due to a large rate of heat exchange between the BP and the anode flow, at the leading order, the stack and flow temperatures coincide. 

The importance of equations 37—39 is that once the heat-exchanger effectiveness, S, is known for a given heat exchanger, one can compute the actual heat-transfer rate and outlet stream temperatures from specified inlet conditions. This process is known as rating a given heat exchanger. 

Double-pipe exchangers are often piped in complex series-parallel arrangements on both sides. The MTD to be used has been derived for some of these arrangements and is reported in Kern (Process Heat Transfer, McGraw-Hill, New York, 1950). More complex cases may require trial-and-error balancing of the heat loads ana rate equations for subsections or even for individual exchangers in the bank. 

The theoretical steam rate (sometimes referred to as the water rate) for stream turbines can be determined from Keenan and Keyes or Mollier charts following a constant entropy path. The theoretical steam rate is given as Ib/hr/kw which is easily converted to Ib/hr/hp. One word of caution—in using Keenan and Keyes, steam pressures are given in PSIG. Sea level is the basis. For low steam pressures at high altitudes appropriate coirections must be made. See the section on Pressure Drop Air-Cooled Air Side Heat Exchangers, in this handbook, for the equation to correct atmospheric pressure for altitude. 

Conservation is a general concept widely used in chemical engineering systems analysis. Normally it relates to accounting for flows of heat, mass or momentum (mainly fluid flow) through control volumes within vessels and pipes. This leads to the formation of conservation equations, which, when coupled with the appropriate rate process (for heat, mass or momentum flux respectively), enables equipment (such as heat exchangers, absorbers and pipes etc.) to be sized and its performance in operation predicted. In analysing crystallization and other particulate systems, however, a further conservation equation is... 

Convection is heat transfer between portions of a fluid existing under a thermal gradient. The rate of convection heat transfer is often slow for natural or free convection to rapid for forced convection when artificial means are used to mix or agitate the fluid. The basic equation for designing heat exchangers is... 

The heat balance equation can now be applied to segment n, of the heat exchanger. The heat transfer rate equations are given by the following terms... 

The cost of a carbon steel heat exchanger needs to be adjusted for the material of construction. Because of the low pressure rating, no correction for pressure is required (Table 2.5), but the cost needs to be adjusted for the material of construction. From Table 2.4, /m =2.9, and the total cost of the installed equipment can be estimated from Equation 2.6 and Table 2.7. If the project is a complete new plant, the contribution of the heat exchanger to the total cost can be estimated to be ... 

Process simulators contain the model of the process and thus contain the bulk of the constraints in an optimization problem. The equality constraints ( hard constraints ) include all the mathematical relations that constitute the material and energy balances, the rate equations, the phase relations, the controls, connecting variables, and methods of computing the physical properties used in any of the relations in the model. The inequality constraints ( soft constraints ) include material flow limits maximum heat exchanger areas pressure, temperature, and concentration upper and lower bounds environmental stipulations vessel hold-ups safety constraints and so on. A module is a model of an individual element in a flowsheet (e.g., a reactor) that can be coded, analyzed, debugged, and interpreted by itself. Examine Figure 15.3a and b. 

Emmitle 4.2. A countercurrent heat exchanger is an important example of a system described by equations that are usually solved iteratively. Figure 4.4 shows the system. The problem is to And the steadystate outlet temperatures of the oil, T , and cooling water,, and the heat transfer rate Q, given the inlet temperatures, flow rates, and heat transfer coefficient and area. The steadystate equations for heat transfer are... 

In this equation, U is the heat transfer coefficient in ener per area A, per temperature difference (don t confuse this U with the internal energy U), and A, is the area across which heat exchange occurs between the reactor at temperature T and coolant at temperature We want to define Q as the rate of heat removal as a positive quantity so Q will be positive if r > Tc- If F heat flows into the reactor so the reactor is being heated. We could use Th as the heating temperature, but since the cooled reactor is the more interesting situation, we shall use as the temperature of the fluid, which is exchanging heat with the reactor. 

Equation (293) cannot be applied to gas fluidized beds because in the latter case, the fluidized bed contains a large number of bubbles. The rate of heat transfer between the bed and wall is determined in the latter case by the heat transfer in the packets (clusters) of solid particles (through which the gas flows at the minimum fluidization velocity) which are exchanged, because of bubbling, between the wall and the bulk of the fluidized bed [74], The heat transfer coefficient is given in the latter case by an expression similar to Eq. (282) ... 

Standards have been developed by the Heat Exchange Institute for rates of air leakage into commercially tight systems. Their chart is represented by the equation... 

The process design of double-pipe exchangers is practically the simplest heat exchanger problem. Pressure drop calculation is straightforward. Heat transfer coefficients in annular spaces have been investigated and equations are rated in Table 8.10. A chapter is devoted to this equipment by Kern (1950). 

Consider again the temperature control system fitted to the heat exchanger in Fig. 7.1. Suppose that the temperature of the cold stream decreases. Then, clearly, the temperature at Y, i.e. 0, will also begin to fall. In response to this the controller will open the control valve further in proportion to the error—where the error is given by equation 7.1. In order to maintain this new steady state, i.e. with the increased rate of flow of the hot stream, a constant additional output must be applied to the control valve by the controller. This additional output can exist only if there is an... 

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