Solving heat transfer equation iteratively with energy balance is the most common thing in process engineering. Solving LMTD equation is must in every rigorous heat transfer calculation. It is always easy to use e – NTU method to solve bulk of a heat transfer equipment than using LMTD equation to solve it part by part. LMTD equations diverge if the initial guess values are poor and requires high computational power for accuracy.
This discussion is aimed at creating an awareness to a process engineer who might fumble if fallen in a situation mentioned in the title. Before the core of the concept is disclosed some terms will be explained to make a novice comfortable.
Pinch point: The point of closest approach between the hot and cold composite curves is the pinch point [Wikipedia].
LMTD: Log Mean Temperature difference. This term indicates the non-linearity in the hot and cold temperature profile.
Equation Q = U x A x LMTD is the heat transfer equation which is solved along with energy balance equation to estimate the temperature profile.
Energy Balance equation: Q = W x Cp x (Th -Tc) = M x Cp’ x (th-tc), Where W, Cp, T* belong to hot side and M, Cp’, t* corresponds to cold side.
e-NTU method: It is a method of solving heat transfer equation using the concept of number of transfer units. This technique is useful when both inlet temperatures are available and outlet temperatures are to be estimated.
Let’s get into the Core of Idea:
Consider a case of counter current or cross flow where hot stream is flowing over the tubes and cold fluid within the tubes. If the inlet temperature of hot fluid and outlet temperature of cold fluid is known and the other two are to be evaluated. As mentioned above equations may be modeled and solved to get the unknown temperatures.
A similar attempt is made to evaluate the below mentioned case.
Technical Details:
Solution technique: Secant Method
Problem Variable: Cold fluid inlet temperature
Brief Methodology: Assume the hot fluid outlet temperature. Generate two guess values of cold fluid inlet temperature. Solve the LMTD equation to get a correct cold fluid inlet temperature for the assumed hot fluid outlet temperature. Correct the hot fluid outlet temperature based on energy balance equation. This is a very basic technique and no rocket science involved in it.
Equations are taken in a simple form:
LMTD – K x (Thi- Tho) =0
(Thi – Tho) = N x (tco – tci)
K indicates the ratio of (WCp/UA). M indicates the ratio of (MCp’/WCp).
Equations are solved with constant N value equal to 0.6 and varying the K value from 0.4 to 10.0. 0.4 indicates a case where the area of heat transfer is very high and 10 indicates a lower heat transfer area.Thi and tco are known values and (Thi-tco) is the pinch point in our case.
Thi = 549.15 K
tco = 544.15 K
Note: 5 K pinch is selected.
temperature profile with small pinch (5 K)
Hot fluid outlet temperature and cold fluid inlet temperature difference (toward the extreme right of the above graph) will be referred as end B while pinch end will be referred as end A.
Equations are solved at various K values (0.4, 0.6, 0.8, 1, 10) and different pinch temperatures (starting from 5K to 13K with a unit increment). Equations are solved with a tight tolerance of e = 0.000000001 (1E-9).
What is Interesting in this?
Results obtained by solving above equations at mentioned pinch, K and N values are presented below.
It is clearly observed that the change in pinch from 5 to 6 will deviate the calculated cold stream inlet temperature by 21 K. This non – linearity can be appreciated if the pinch value is estimated with certain error. Any error in the estimated pinch value (even by 1 K) can upset your results drastically. To quantify exactly if you compare case where K=0.4 and K=10.0 (high heat transfer area and low heat transfer area cases) a 9K difference in pinch will deviate results by 44.6% & 0.3% respectively in prediction of cold stream inlet temperature. If started with K=10 and carried out the calculations in series (estimated values for i segment will be inputs for i+1 segment and i=1 to 25) for 25 times an equally high deviation as 44.6% will be seen. [Idea is solving bulk heat transfer area and section wise heat transfer area]
If we analyze how the temperature difference at pinch end (end A) can change the temperature estimates at end B (see the above graph), it is clear that the temperature difference at other end changes 10.14 times faster than the temperature difference at pinch end.
Finally What is to be done and what not?
Carrying out solution for LMTD equation to find the temperatures at same end of a counter current flow given the other end temperatures is not always a good idea (when end A is given and end B is to e estimated). If the pinch side temperatures are accurate, then the model will work. But, if the values are approximate or obtained from a model then this method is not suggested. Any error in the pinch side will create a huge difference in the end B. As a preliminary estimate e-NTU method may be applied to check the validity of the heat transfer coefficients obtained from theory and proceed for LMTD method for rigorous calculation. It can be concluded that while calculation moving from the pinch end (end A) to the end B may give erratic results.
So, this method should be applied with caution and proper checks.
