Heat capacity rate

Assuming that the maximum possible temperature difference is on the fluid side with the higher heat capacity rate, then... 

If the cold and hot fluids heat capacity rates are equal, then R = 1. Equation (9.22) gives an indefinite value, and this equation cannot be used directly. Using I Hopital s rule as R —> 1 gives d... 

Calculating the heat capacity rates ratio gives... 

To determine the true overall temperature difference, the correction factors, F, shown in Figure 10-34 are used to correct for the deviations involved in the construction of multipasses on the shell and tube sides of the exchanger. Note that R of the charts represents the heat capacity rate ratio , and P is the temperature efficiency of the exchanger. 

In heat exchanger analysis, it is often convenient to combine the product of the mass flow rale and the specific heat of a fluid into a single quantity. This quantity is called the heat capacity rate and is defined for the hot and cold fluid streams as... 

With the definition of the heat capacity rate above, Eqs. 11-9 and 11-10 cau also be expressed as... 

That is, the heat transfer rate in a heat exchanger is equal to the heat capacity rate of cither fluid multiplied by the temperature change of that fluid. Note that /ie only lime the temperature rise of a cold fluid is equal to the temperature drop of the hot fluid is Vi hen the heat capacity rales of the two fluids are equal to each other (Fig. 11-12). 

The heat transfer in a heat exchanger will reach its maximum value when (1) the cold fluid is heated to the inlet temperature of the hot fluid or (2) the hot fluid is cooled to the inlet temperature of the cold fluid. These two limiting conditions will not be reached simultaneously unless the heat capacity rates of the hot and cold fluids are identical (i.e., Q = Q). When Q = which is usually the case, the fluid with the smaller heat capacity rate will experience a larger temperature change, and thus it will be the first to experience the maximum temperature, at which point the heat transfer will come to a halt. Therefore, the maximum possible heat transfer rate in a heat exchanger i.s (Pig. n-23)... 

Properties The specific heat of water is given to be Cp = 4.18 kJ/kg °C. Analysis A schematic of the heat exchanger is given in Fig. 11-24. The heat capacity rates of the hot and cold fluids are... 

You may be tempted to think lhat the cold water should be heated to 70 C in the limiting case of maximum heat transfer. But this will require the temperature of the. hot water to drop to -170 C (below 10°C), which is impossible. Therefore, heat transfer in a heat exchanger reaches Its maximum value when (he fluid with the smaller heat capacity rate (or the smaller mass flow rate when both fluids have the same specific heat value) experiences the maximum temperature change. This example explains why vie use in the evaluation of instead of... 

Our objective is to develop relationships between the heat transfer rate q exchanged between two fluids, heat transfer surface area A, heat capacity rates C of individual fluid streams, overall heat transfer coefficient U, and fluid terminal temperatures. In this section, starting with idealizations for heat exchanger analysis and the thermal circuit associated with a two-fluid exchanger, e-NTU, P-NTU, and MTD methods used for an exchanger analysis are presented,... 

Here dq is heat transfer rate from the hot to cold fluid across the surface area dA C and Cc are the heat capacity rates for the hot and cold fluids, and the sign depends on whether dTc is increasing or decreasing with increasing dA. The overall rate equation on a local basis is... 

In all formulas of plate heat exchangers with the number of thermal plates N — (equation numbers starting with V), the single-pass parallelflow and counterflow temperature effectivenesses are presented in implicit forms. Their explicit forms are as follows, with x and y representing the appropriate values of the number of transfer units and heat capacity rate ratios, respectively. 

In order to determine the j factor on the unknown side, the exchanger effectiveness e is determined from the temperature measurements, and the heat capacity rate ratio C is determined from individual flow measurements and specific heats. NTU is subsequently computed from the appropriate e-NTU relationship for the test core flow arrangement (such as Eq. II. 1 in Table 17.6). Generally, the test section is a new exchanger core, and fouling resistances are negligible T 0hA on the unknown side is determined from the following thermal resistance equation where UA is found from NTU ... 

From the known heat capacity rates on each fluid side, compute C = Cmm/Cmax. From the known UA, determine NTU = fM/Cmin. Also calculate the longitudinal conduction parameter X. With the known NTU, C, X, and the flow arrangement, determine the crossflow exchanger effectiveness (from either closed-form equations of Table 17.6 or tabular/ graphical results from Kays and London [20]. 

The sum of the heat capacity rates for all smaller units is equal to the total heat capacity rates of the real maldistributed heat exchanger. 

The method is proposed only for the counterflow, parallelflow and crossflow arrangement with one fluid unmixed throughout. Analytical expressions for temperature effectivenesses are provided in Table 17.6 for the uniform flow case and are used for individual N hypothetical units. The heat capacity rate of the maldistributed fluid in the nth subheat exchanger, as in a counterflow arrangement, is given by... 

CJC, = 1 is the worst case at large NTUs (NTUs based on the shell fluid heat capacity rate). 

C Flow stream heat capacity rate with a subscript cot h, rhCp, W/°C, Btu/hr °F... 

C Heat capacity rate ratio, Cmln/Cmax, dimensionless... 

Cn, Heat capacity rate of the maldistributed stream, W/°C, Btu/hr °F Co Convection number (defined in Table 17.26), dimensionless... 

Cr Heat capacity rate of a regenerator, M c N or Mwcwl<3 , for the hot and cold side... 

C Total matrix heat capacity rate ratio, Cr/Cmin, C% = Crj,ICh, C% = CrclCc,... 

Number of heat transfer units based on fluid 1 heat capacity rate, UAIC similarly, NTU2 = UAIC2, dimensionless... 

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