게시판 > 질답게시판 > k=Cp/Cv, Specific Heat ratio에 대한 질문입니다.

공룡	11-01-06 07:37

도움이 될지 모른다는 생각에 적습니다. (저는 HYSYS Dynamics 개발 업무를 하는 엔지니어입니다.)
지금 어떤 HYSYS 버젼을 사용하고 계신지는 잘 모르지만, HYSYS에서 k 값의 계산에 중대한 오류가 발견되어 최근 수정하였습니다.

API-520 과 ANSI/ISA–75.01.01–2002 (IEC 60534-2-1) 표준 정의에 따르면, k 값의 정의는 1atm 을 기준으로 하여 만들어진 것입니다. HYSYS 에서 발견된 오류는 Cp/(Cp-R) 에서 Cp 계산을 1 atm 이 아닌 실제압력에서 했다는 것이었어요. 따라서, 나열하신 k 값은 계산 오류가 포함된 것일거에요.

조만간 출시되는 V7.3 에 implement 되었고, V7.1 에도 patch를 만들어 support 로 보낸것으로 기억합니다. (제가 한건데도 기억이 가물가물..). 정확히 compressor 에도 해당이 되는지 확인하지 않고 말씀드리는 것이니 착오가 있을련지도 모르겠습니다만, HYSYS fluid package 의 method 들이 기본적으로 common 이므로 아마도 해당이 될거라는 추측을 합니다.

추가하여, 그 계산에서 두번째 식 (Cp/Cv) 와 세번째 식은 Cp 계산이 1atm 에서 된것이 아니니 사용하시면 안될 것 같고요. V7.3 부터 첫번째 식만 유효할 것 같네요.

어쩌면, 원하시는 답이 아닐지도 모르지만 혹시 도움이 될까하여 적었습니다.

공룡 dinoboy

11-01-06 07:53

What's New in V7.3 를 위해 만들었던 내용이 있어 카피합니다.

What’s New in HYSYS Dynamics V7.3
Ratio of specific heats calculation by Semi-Ideal Cp/Cv method

In the Relief valve, the critical flow pressure ratio is determined by using the ideal gas relationship given in Eqn (6.46) of HYSYS Unit Operations Guide:

P2/P1 = [2/(K+1)]^(K/(K+1)) ............... (eqn 6.46)

K is the ratio of specific heats, and it is calculated by either <Rigorous Cp/Cv> or <Semi-Ideal Cp/Cv> methods. In V7.3 onwards, when <Semi-Ideal Cp/Cv> is selected, the specific heat ratio factor is calculated by atmospheric pressure (1 atm), while the earlier HYSYS versions use the process gas pressures. This change could cause differences in the result, when the process pressure is different to the atmospheric pressure.

Semi-Ideal Cp/Cv option for ISA standard valve sizing

A new option for ANSI/ISA standard valve sizing has been implemented. On the Rating tab, Sizing (dynamics) view of Valve model, there is new option allowing the user to select between <Rigorous Cp/Cv> and <Semi-Ideal Cp/Cv> methods. This option determines how mixture’s heat capacity is being calculated therefore the specific heat ratio factor [reference: ANSI/ISA–75.01.01–2002 (IEC 60534-2-1 Mod), page 23] is evaluated accordingly. When <Semi-Ideal Cp/Cv> is selected, the specific heat ratio factor is calculated using the approximation, Cv = Cp - R , and Cp is the heat capacity of the process mixture at the atmospheric pressure (1 atm). This new method meets the ANSI/ISA standard and also meets the API 520 standard, for relief and control valves at choked flow.
Meanwhile, <Rigorous Cp/Cv> is the original HYSYS method, and it uses the process pressure condition, which is dependent on pressures and will then deviate from ANSI/ISA standard. Due to compatibility reason, <Rigorous Cp/Cv> is the default method in the Valve model.

What's New in V7.3 를 위해 만들었던 내용이 있어 카피합니다.

What’s New in HYSYS Dynamics V7.3
Ratio of specific heats calculation by Semi-Ideal Cp/Cv method

In the Relief valve, the critical flow pressure ratio is determined by using the ideal gas relationship given in Eqn (6.46) of HYSYS Unit Operations Guide:

P2/P1 = [2/(K+1)]^(K/(K+1)) ............... (eqn 6.46)
 
K is the ratio of specific heats, and it is calculated by either <Rigorous Cp/Cv> or <Semi-Ideal Cp/Cv> methods. In V7.3 onwards, when <Semi-Ideal Cp/Cv> is selected, the specific heat ratio factor is calculated by atmospheric pressure (1 atm), while the earlier HYSYS versions use the process gas pressures. This change could cause differences in the result, when the process pressure is different to the atmospheric pressure.

Semi-Ideal Cp/Cv option for ISA standard valve sizing

A new option for ANSI/ISA standard valve sizing has been implemented. On the Rating tab, Sizing (dynamics) view of Valve model, there is new option allowing the user to select between <Rigorous Cp/Cv> and <Semi-Ideal Cp/Cv> methods. This option determines how mixture’s heat capacity is being calculated therefore the specific heat ratio factor [reference: ANSI/ISA–75.01.01–2002 (IEC 60534-2-1 Mod), page 23] is evaluated accordingly. When <Semi-Ideal Cp/Cv> is selected, the specific heat ratio factor is calculated using the approximation, Cv = Cp - R , and Cp is the heat capacity of the process mixture at the atmospheric pressure (1 atm). This new method meets the ANSI/ISA standard and also meets the API 520 standard, for relief and control valves at choked flow.
Meanwhile, <Rigorous Cp/Cv> is the original HYSYS method, and it uses the process pressure condition, which is dependent on pressures and will then deviate from ANSI/ISA standard. Due to compatibility reason, <Rigorous Cp/Cv> is the default method in the Valve model.

장현수 jddast

11-01-06 10:38

우선 공룡님 감사합니다. 진짜 큰 도움이 되었습니다.
API 520확인결과 PSV에 적용되는 k값은 확실히 1atm and Relieving Temperature에서 값이라고 명시가 되어있네요. Handbook에서 찾은 Ethylene의 1atm에서의 k값확인 결과 1.243(at average temperatures of suc. & disch. and 1atm)으로 제가 최초질문2번에서 HYSYS가 계산한 k값과 일치함을 확인했습니다.

추가적인 궁금증으로
(1) 위에서 언급했던 HYSYS에서 나오는 3가지 k값은 각각 어디에 해당이 되는건지요?
http://www.jmcampbell.com/may-2009.php Campbell사 자료에 의하면 compressor에서 적용되는 k값의 정의는 6가지가 있더군요.

1) k at Suction Conditions
2) k at Discharge Conditions
3) k at T_average and 1atm
4) k at T_average and P_average
5) 1번k와 2번k의 평균
6) 1번k와 2번k, 중간압력k의 평균

제판단으로 첫번째 k는 3번, 두번째와 세번째는 잘 모르겠네요. 특히 세번째에 Ent. Method라는게 무엇인지 궁금합니다.

(2) Compressor Performance창에 Isentropic Exponent라는 항목의 값은 또 위에 3가지 k값과 다르더라구요. 이것도 오류에 의한 것인지요?

공룡님께서 말씀하신 "HYSYS 에서 발견된 오류는 Cp/(Cp-R) 에서 Cp 계산을 1 atm 이 아닌 실제압력에서 했다는 것이었어요. 따라서, 나열하신 k 값은 계산 오류가 포함된 것일거에요. " 는 3번으로 계산되어야 했으나 4번으로 계산이 잘못되어 오류를 수정하셨다는 말씀이라 판단이 되네요. 덕분에 공부를 많이 하고 있는것 같습니다. 질문이 많아 죄송하지만 정말 알고 싶어 그렇습니다. 감사합니다.

공룡 dinoboy

11-01-07 01:59

(1) 첫번째 K 는 Discharge T와 P=1atm 으로 계산된 Cp 값과 기체상수 이용해서 계산하고 (오류수정 이전은 P=discharge P), 두번째는 Discharge T&P 에서 Cp/Cv 계산된 것이고, 세번째는 (구체적으로 들여다 보진 않았으나) 변수명으로 보아 PH 플래시 계산으로부터 산출한 Cp/Cv 로 계산한 것일것 같습니다. (그리 중요한 것은 아닌듯..)

(2) Isentropic exponent 계산은 runtime 에서 두가지 다른방법으로 산출하는데, 첫번째 경우는 두번째 k 값과 동일한 값을 보이거나, 만일 해당조건 (proprietary 한 것 같아.. 구체 설명은 않겠습니다.. 죄송..) 이 맞지 않다면 isentropic 플래시 계산 (PS flash) 구해진 density과 Feed density 비율과 feed와 discharge 간의 압력비를 이용, 아래 계산식을 적용하고 있네요.

densityratio = (1/Rho_isen) / FeedDensity
pressureratio = ProductPressure/FeedPressure
IsentropicExponent = Log(pressureratio)/Log(densityratio)

장현수 jddast

11-01-07 16:26

친절한 답변 감사합니다. Isentropic exponent의 식에 대해서는 따로 조사를 더 해봐야겠네요.
감사합니다.

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