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# Method of Calculating Heat by Using Constant Pressure Heat Capacity and Its Application in Heat Exch

Heat exchanger is widely used in industrial production and daily life. A reasonably designed heat exchanger should not only meet the needs of process and daily life, but also achieve the purposes of energy saving, consumption reduction and improving production efficiency. The most common in industrial production is the wall heat exchanger. The design basis of wall heat exchanger is q = Ka Î” TM, where Q - heat transfer rate, numerically equal to heat load, w or kW, K - heat transfer coefficient, w / m2k, Î” TM - average temperature difference K. The calculation of heat load Q in the formula is very important. Here, only the correct application of constant pressure heat capacity in heat load calculation is discussed for reference.

1ã€ Heat capacity of matter

The sensible heat required for a certain amount of material to rise 1K is called heat capacity. The heat capacity of matter is related to the heating conditions. The heat capacity is divided into constant pressure heat capacity and constant volume heat capacity. The heat required for a certain amount of a substance to increase its temperature by 1K at constant pressure is the constant pressure heat capacity CP. The heat required for a certain amount of a substance to increase its temperature by 1K at a constant volume is the constant volume heat capacity cv. Usually, the heat capacity of a substance increases with the increase of temperature. Since most chemical processes are carried out under constant pressure, only constant pressure heat capacity is introduced below.

1. True constant pressure heat capacity (hereinafter referred to as true heat capacity CP)

The true heat capacity of a substance at a certain temperature is defined as: CP = DQP / dt = (9h / 9t) p the effect of temperature on the heat capacity of various substances is usually expressed as CP = f (T), and the common ones are CP = a Bt CT2, CP = a Bt C â€² T22. In the formula, a, B, C and C â€² are the characteristic constants of various substances measured experimentally (generally available in the chemical and chemical engineering manuals), and t is the absolute temperature, That is, QP = âˆ« t2-t1cpdt. It is more accurate to calculate the constant pressure heat QP with this functional relationship.

2. Average constant pressure heat capacity (hereinafter referred to as average heat capacity CP)

CP is commonly used in engineering calculation to calculate QP. If heat QP is required when n mole of a substance rises from T1 to T2 under constant pressure, the average heat capacity of the substance within the temperature range can be defined as: CP = QP / N (t2-2t1) = âˆ« t2-t1cpdt / (T2 - T1), that is, the heat required for an average temperature rise of 1K within the temperature range of T1 T2 of a mole of substance. With the average heat capacity, the constant pressure heat QP can be obtained from QP = n CP (t2-t1). The value of CP is strictly related to the temperature range.

2ã€ Method of calculating heat by constant pressure heat capacity

There are many methods to calculate heat (i.e. constant pressure heat) with constant pressure heat capacity. The following introduces several common methods by taking the heat calculation of 1kmolco2 from 100 â„ƒ 600 â„ƒ under normal pressure as an example.

1. True heat capacity method

(1) True heat capacity integration method (method 1)

Qp = nâˆ«T2-T1CpdT = nâˆ«T2-T1ï¼ˆa bT cT2ï¼‰ dT

Where: QP - heat change of material under constant pressure, kJ;

N - molar number of substance, KMOL;

CP - constant pressure true heat capacity of material kJ / KMOL. K;

T1, T2 - temperature K of substance in initial and final state;

According to the manual, a = 26.75, B = 42.258 for CO2 Ã— 10-2ã€c = 14. 25 Ã— 10-6 i.e

Qp =âˆ«873. 15-373. 15ï¼ˆ26. 75 42. 258 Ã— 10-22- 14. 25 Ã— 10-6T2 ï¼‰ dT = 23626K

(2) Check the true heat capacity at the average temperature (method 2)

Qp = nCp ï¼ˆ T2 - T1ï¼‰

Where: CP - true heat capacity of substance at (T1 T2) / 2 kJ / KMOL. K

According to the manual, CP = 47. 76kj / KMOL. K at co2350 â„ƒ,

Qp = 1 Ã— 47. 6 Ã— ï¼ˆ6002100ï¼‰ = 23 880KJ

2. Average heat capacity method

(1) Check the average heat capacity under the initial and final temperature (method 3)

Qp = n Cp2t2 - n Cp1t1

Where: CP1, CP2 - average heat capacity of initial and final states kJ / KMOL. K;

T1, T2 - temperature in initial and final state Â° C.

It is found that the average heat capacity of CO2 at 100 â„ƒ and 600 â„ƒ is 39.15 and 45.43kj / KMOL. K respectively,

Qp = 1 Ã— 45. 43 Ã— 600 - 1 Ã— 39. 15 Ã— 100 = 23 343KJ

(2) Check the average heat capacity at the average temperature (method 4)

Qp = n Cp ï¼ˆT2 - T1ï¼‰

Where: CP - average heat capacity of substance at (T1 T2) / 2 kJ / KMOL. K.

It is found that the average heat capacity of CO2 at 350 â„ƒ is 42.50kj / KMOL. K

Qp = 1 Ã— 42. 50 Ã— 500 = 21 250KJ

3. Analysis and discussion of calculation results

(1) Summary of calculation results

The error in the table refers to the error value of each method compared with method 1.

(2) Comparison of various methods

In Table 1, each method is compared based on Method 1 because this method has a strict theoretical basis and accurate calculation results. However, this method requires integration, which is very troublesome and rarely used in engineering calculation.

The error of method 2 is small, but the scope of application is limited. It is only suitable for the case where CP = f (T) is a linear relationship. When CP = f (T) is a linear relationship, the relationship between the average heat capacity and the true heat capacity is CP = (CP1 CP2) / 2. At this time, the average heat capacity is the true heat capacity at the average temperature (T1 T2) / 2.

The error of method 3 is small. For some gases that are difficult to find the true heat capacity, the average heat capacity (0 t â„ƒ) can be checked at two temperatures for heat calculation.

The processing method of method 4 is wrong and the calculation result is too small. When calculating the constant pressure heat QP from QP = n CP (T2 - T1), the value of CP is strictly related to the temperature range and must be the average heat capacity in the range of T1 T2. The average heat capacity of some common gases in the range of 0 t â„ƒ is provided in the general manual. The average heat capacity used in the example is 42.50kj/kmol. K is the average heat capacity in the range of 0 350 â„ƒ, which is fundamentally different from the temperature range of 100 â„ƒ 600 â„ƒ in the calculation formula and cannot be applied. This is a problem that is often easily overlooked by some engineers and textbooks, and should be paid attention to.

4. Influence of different selection of CP on heat transfer area of heat exchanger

For example, a tubular heat exchanger is designed to cool 52 700kg of benzene per hour from 353.1k to 308k with 303k water (outlet temperature is 310K). Benzene is selected to go through the tube side. It is known that the heat transfer coefficient k = 493w / m2. K. Average temperature difference = 14.3k, calculate the heat transfer area with methods 4 and 3 as follows:

Method 4: Q1 = w CP (T1 - T2) = 52 700 Ã— 1. 84 Ã— ï¼ˆ353. 1 - 308ï¼‰= 4. 4 Ã— 106KJ / h

Heat transfer area a = Q1 / K Î” tm = 4. 4 Ã— one hundred and six Ã— 103/ 493 Ã— 14. 3 Ã— 3 600= 177m2

Considering a safety factor of 15%, the actual area is 177 Ã— 1. 15 = 199m2

Method 3: Q2 = w [CP1 (T1 - t0) - CP2 (T2 - t0)] = 52700 Ã—ï¼» 1. 91 Ã— ï¼ˆ352. 1 - 273ï¼‰ - 1. 79 Ã— ï¼ˆ308 - 273ï¼‰ ï¼½= 4. 75 Ã— 106KJ / h

Heat transfer area a = Q2 / K Î” tm = 4. 75 Ã— one hundred and six Ã— 103/ 493 Ã— 14. 3 Ã— 3 600= 187m2

Considering the safety factor of 15%, the actual area is 187 Ã— 1. 15 = 215m2

The above calculation further shows that the method of checking the average heat capacity at two temperatures is often used in the design of heat exchanger, because this method meets the temperature range required by the formula.

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