NTU Method of Performance Evaluation (Number of Transfer Units)
NTU Method of Performance Evaluation (Number of Transfer Units)
The NTU method is more elegant. Textbooks on heat transfer will explain the basis of this method for various arrangements of the heat transfer surface such as counter-flow, parallel-flow, single- or multi-pass cross-flow, and so on [3–5].
Using this method, one can estimate the duty of heat transfer equipment with no phase change (and with little or no external radiation) in one step, given the flue gas and fluid
Appendix A: Boiler Design and Performance Calculations 367
inlet temperatures, surface area, and U. Though we say it is one-step method, it is in reality not so as thermal and transport properties used in the heat transfer evaluation depend on the average gas and fluid temperatures and also U. When one has only the inlet tempera- ture of both the fluids, the assumption of gas properties cannot be accurate. An evaluation has to be made of the average gas and steam or water temperature and the specific heats, and U may be fine-tuned after a few runs. With a computer program, one may achieve the final results in short order. In boiler practice, we use it for predicting the performance of a superheater or economizer or the air heater. A slightly different procedure is used for an evaporator with phase change at constant temperature, which will be discussed later. The steps are shown in the following text for clarity.
The duty of any heat transfer equipment is given by
Q =∈ C min ( T 1 − t 1 ) (A.8)
where ∈ is the effectiveness factor, which depends on the heat exchanger configuration (parallel- or counter-flow or multi-pass cross-flow, etc.) See Table A.2 for ∈ values for
some configurations. T 1 ,t 1 are inlet gas and inlet fluid temperatures, °C.
( WC
p min ) = C min (A.9)
( WC p max ) C max
and
C min =
( WC
p ) min (A.10a)
C max = ( WC
) (A.10b)
p max
TABLE A.2
Effectiveness Factors for Exchangers
Exchanger Type
Effectiveness ∈
Parallel-flow, single-pass 1 ε = − exp[ − NTU × ( 1 + C )] 1 + C
1 − exp[ − NTU × ( 1 − C )] Counter-flow, single-pass
− C exp[ − NTU × ( 1 − C )]
++ C + exp[ − NTU × ( 1 + C 212 )] / 21 / /2
Shell-and-tube (one shell pass; 2, 4, 6, etc., tube passes)
×+ ( 1 C )
1 − exp[ − NTU ×+ ( 1 C )]
C Shell-and-tube (n shell pass; 2n, 4n, 6n, etc., tube passes)
Cross-flow, both streams unmixed ε ≈ 1 − exp{ C × NTU . 0 22 [exp( − C × NTU . 0 78 ) − 1 ]}
Cross-flow, both streams mixed
ε NT U
NT U
NT U × C = + − 1 1 − exp( − NT U ) 1 − exp( − NT U × C )
Cross-flow, streams C min unmixed ε =− { 1 exp[ − C [ 1 − exp( − NTU )]]} / C Cross-flow, streams C max unmixed
ε = 1 − exp{ − 1 / C [ 1 − exp( − NTU × C )]}
368 Appendix A: Boiler Design and Performance Calculations
(WC p (mass flow × specific heat) of the flue gas and the other fluid such as steam or water or air is computed, and the lower of the two values is (WC p ) min . Typically, in a steam genera- tor, the tube side (WC p ) will be larger while in gas turbine exhaust boilers (HRSGs), the gas side (WC p ) will be larger.)
NTU is the number of transfer units =
( (A.11)
Note that NTU is nondimensional. Hence, we can use kcal/m 2 h °C (Btu/ft 2 h °F)(W/m 2 K) for U, m 2 (ft 2 ) for A, kg/h (kg/s) (lb/h) for W, kcal/kg °C (kJ/kg K) (Btu/lb °F) for C p . Using the effectiveness factor from Table A.2, one computes the duty and then arrives at the exit flue gas and fluid temperatures. One may note that though NTU method appears to be a direct calculation procedure for the duty and exit temperatures of a heat transfer equipment, a few iterations are required to perform this calculation accurately as U and specific heats of hot and cold fluids vary with the average gas and fluid temperatures, and the average gas and fluid temperatures depend on the inlet and exit temperatures of both the fluids! A well-written program is helpful.
Example A.2
70,000 kg/h of flue gases with the same analysis as in Example A.1 enters this superheater
at 500°C while 19,000 kg/h of saturated steam flows inside the tubes at 51 kg/cm 2 a. What
will be the duty and the flue gas and steam exit temperatures? Assume that the flue gas average specific heat is 0.273 kcal/kg °C and that of steam 0.768 kcal/kg °C (specific heat is obtained by dividing the enthalpy difference by the temperature difference. These may also be checked later based on actual temperatures if they differ significantly from
those assumed.) Surface area is 188 m 2 .
Solution
Flue gas WC p = 70,000 × 0.273 = 19,110 and for steam, WC p = 19,000 × 0.768 = 14,592. Hence, (WC p ) min is 14,592. (Heat loss factor on gas side is neglected here.) From Table A.2 for counter-flow arrangement,
∈= [ 1 − exp{ NTU − ×− ( 1 C )} ] (A.12a)
C exp{ NTU − ×− ( 1 C ) }]
where
NT U = UA and C = ( ) min
WC p
C min
WC p ( ) max
( WC p ) min =
U for the full load case was 51.9 kcal/m 2 h °C (10.6 Btu/ft 2 h °F), and for this case, we may
assume that U varies as a function of gas flow to the power of 0.65 and use an approxi-
mate value of 51.9 × (70,000/100,000) 0.65 = 41.1 kcal/m 2 h °C (8.42 Btu/ft 2 h °F). This can be
Appendix A: Boiler Design and Performance Calculations 369
checked later if a computer program is used. We made use of the fact that gas-side heat transfer coefficient is governing U. NTU = 41.1 × 188/14,592 = 0.53.
Hence, Q = 0.36 × 14,592 × (500 − 264) = 1.24 × 10 6 kcal/h (1,442 kW) (4.92 MM Btu/h) Exit gas temperature = 500 1 24 10 70000 273 − 6 . × / / . = 435 ° C 815 ° F
Exit steam temperature = 264 1 24 10 19 000 0 768 349 + . × 6 /, / . = ° C 660 ° F F
These are good estimates. With a computer program, one may fine-tune the values of U, C p of steam and flue gas and estimate the duty more accurately. The procedure explains how off-design calculation may be done for an existing equipment. One can then proceed to check the gas- and steam-side pressure drops and tube and fin tip tem- peratures as detailed in Appendix E.
One may also check these results as follows. Compute ΔT = [(500 − 349) − (435 − 264)]/ ln[(500 − 349(/(435 − 264)] = 160.7°C. Q t = UAΔT = 41.1 × 188 × 160.7 = 1.242 MM kcal/h.