In this article we will discuss about the performance of heat exchangers.
The indicated parameters are:
m = mass flow (kg/ s)
c = specific heat (J/kg-deg)
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t = fluid temperature (deg C)
Δt = temperature drop or rise of a fluid across the heat exchanger.
Subscripts h and c designate the hot and cold fluids respectively; subscripts 1 and 2 correspond to the inlet and outlet conditions of the fluid.
The following aspects are considered individually in the design and performance analysis of a heat exchanger:
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(i) The hot fluid gives up heat
Qh = mh ch (th1 – th2) … (13.1)
(ii) The coolant picks up heat
Qc = mc cc (tc2 – tc1) … (13.2)
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(iii) The structure of the heat exchanger transfers the heat from the hot fluid to the coolant.
Qex = U A ϴm … (13.3)
Where, U is the overall heat transfer coefficient between the two fluids, A is the effective heat transfer area and ϴm is the appropriate mean temperature difference across the heat exchanger structure.
From energy balance, the heat given up by the hot fluid is picked up by the coolant on being transferred through the heat exchanger.
ADVERTISEMENTS:
Qh = Qc = Qex …. (13.4)
Overall Heat Transfer Coefficient:
A heat exchanger is essentially a device in which energy is transferred from one fluid to another across a good conducting solid wall.
Recapitulate that the rate of heat transfer between two fluids is given by:
Apparently the overall coefficient of heat transfer (U) is defined in terms of the total thermal resistance (Rt).
When the two fluids of the heat exchanger are separated by a plane wall, the thermal resistance comprises:
(i) Convection resistance due to the fluid film at the inside surface, 1/Ai hi
(ii) Wall conduction resistance, δ/kA
(iii) Convection resistance due to fluid film at the outside surface, 1/A0 h0.
A plane wall has a constant cross-sectional area normal to the heat flow i.e., Ai = A = A0. That gives-
For a cylindrical separating wall, the cross-sectional area of the heat flow path is not constant but varies with radius. It then becomes necessary to specify the area upon which the overall heat transfer coefficient is based. Thus depending upon whether the inner or outer area is specified, two different values are defined for U.
Equations 13.10 to 13.12 are essentially valid only for clean and uncorroded surface. However, during normal operation the tube surfaces get covered by deposits of ash, soot, dirt and scale etc.
This phenomenon of rust formation and deposition of fluid impurities is called fouling. The surface deposits increase thermal resistance with a corresponding drop in the performance of the heat exchange equipment.
Since the thickness and thermal conductivity of the scale deposits are difficult to ascertain, the effect of scale on heat flow is considered by specifying an equivalent scale heat transfer coefficient, hs. If hsi and hso denote the heat transfer coefficients for the scale formed on the inside and outside surfaces respectively, then the thermal resistance due to scale formation on the inside surface is
and the thermal resistance due to scale formation on the outside surface is –
With the inclusion of these resistances at the inner and outer surfaces,
The reciprocal of scale heat transfer coefficient is called the fouling factor Rf.
Fouling factors are determined experimentally by testing the heat exchanger in both the clean and dirty conditions:
A heat exchanger might be designed either to restrict or to enhance the heat exchange rate. If the heat exchanger is intended to improve heat exchange, U will generally be much greater than 40 W/m2 K. If it is intended to impede heat flow, U will be much less than 10 W/m2 K.
The following points are worth bearing in mind;
(i) The overall heat transfer coefficient depends upon the flow rate and properties of the fluid, the material thickness and surface condition of tubes and the geometrical configuration of the heat exchanger.
(ii) The fluids with low thermal conductivities such as tars, oils or any of the gases usually give low values of heat transfer coefficient, h. The overall coefficient U will generally decrease when such a fluid flows on one side of the exchanger.
(iii) The highly conducting liquids such as water and liquid metals give higher values of heat transfer coefficient h and overall heat transfer coefficient U. Condensation and boiling process also have high values of U.
(iv) For an efficient and effective design, there should be no high thermal resistance in the heat flow path; all the resistance in the heat exchanger must be low.
Logarithmic Mean Temperature Difference:
During heat exchange between two fluids, the temperature of the fluids change in the direction of flow and consequently there occurs a change in the thermal head causing the flow of heat. Fig. 13.10(a) represents the temperature conditions existing in surface condenser or feed water heater.
The hot fluid is steam and the cold fluid is water. Here the temperature of steam remains constant but the temperature of water is progressively rising. During heat exchange in evaporation of water into steam (Fig. 13.10 b), the water evaporates at constant temperature and the temperature of hot gases continuously decreases in flowing from inlet to outlet.
In many instances, both the fluids experience a change in temperature while flowing through the heat exchange equipment. In a parallel flow system, the thermal head (temperature potential) causing the flow of heat is maximum at the inlet. However, the thermal head goes on diminishing along the flow path and is minimum at the outlet. In a counter flow system, both the fluids are in their coldest state at the exit.
To calculate the rate of heat transfer by the expression, Q = U A Δt, an average value of the temperature difference between the fluids has to be determined.
The calculations become simplified if it is assumed that:
(i) The overall heat transfer coefficient U is constant throughout the heat exchanger
(ii) The specific heats and mass flow rates of both the fluids are constant. This also implies that heat capacities (a product of mass and specific heat) of the fluids are constant over the entire length of flow path.
(iii) The exchanger is perfectly insulated and so the heat loss to the surroundings is negligible.
(iv) The temperature at any cross-section of the stream is uniform, i.e., at any cross-section of heat exchanger, each of the fluid can be characterised by a single temperature
(v) There is no conduction of heat along the tubes of heat exchanger
(vi) The kinetic and potential energy changes are negligible
(vii) There is no partial phase change in the systems. The analysis would thus be applicable for sensible heat changes and for the cases when condensation or vaporisation is isothermal over the entire flow path.
Consider heat transfer across an element of length dx at a distance x from the entrance side of the heat exchanger. Let at this section, the temperature of the hot fluid be th and that of cold fluid be tc. Within the limits of this elementary length, the temperature th and t of the heat exchanging fluids can be considered to remain constant.
Then heat flow dQ through this elementary length is given by:
Where, ϴ = (th – tc), the temperature difference between the fluids. Due to heat exchange, the temperature of the hot fluid decreases by dth and the temperature of cold fluid increases by dtc. Then for heat exchange between the fluids, we can write –
Where, Ch = mh ch = heat capacity or water equivalent of the hot fluid
and Cc = mc cc = heat capacity or water equivalent of the cold fluid.
mh and mc are the mass flow rates of fluids and ch and cc are the respective specific heats.
From equation 13.18,
In a counter flow system, the temperatures of both the fluids decrease in the direction of heat exchanger length. In that case-
The +ve sign refers to the parallel flow heat exchanger and -ve sign refers to the counter flow heat exchanger.
Integration of equation 13.21 between the inlet 1 and outlet 2 gives:
Clearly the effective temperature difference must equal 01 and 02 in case the temperature differences on either side of a heat exchanger are equal. Further, the logarithmic mean temperature difference for a counter flow unit is greater than that of a parallel flow system and accordingly the counter flow heat exchanger can transfer more heat than a similar parallel flow heat exchanger. Conversely a counter flow exchanger needs a smaller heating surface for the same rate of heat transfer.
If the variation in the temperature of the fluids is relatively small, then temperature variation curves are approximately straight lines and sufficiently accurate results are obtained by taking the arithmetic mean temperature difference (AMTD).
Practical consideration, however, suggest that at ratios ϴ1/ϴ2 > 1.7, the logarithmic mean temperature difference should be invariably used.
Effectiveness and Number of Transfer Units (NTU):
The concept of log-mean temperature difference for estimating/analysing the performance of a heat exchanger unit is quite useful only when the inlet and outlet temperatures of the fluids are either known or can be easily determined from the relevant data.
The usual design is, however, based on known fluid inlet temperature and estimated heat transfer coefficients. The unknown parameters may be the outlet conditions and heat transfer or the surface area required for a specified heat transfer. From energy balance, the heat given up by the hot fluid is picked up by the coolant on being transferred through the heat exchanger. Thus,
The outlet conditions for each fluid can be worked out by eliminating Q between the above equations. The resulting equation however becomes unwieldy requiring a trial-and-error iteration approach owing to the logarithmic function in ϴm.
An estimate/analysis of the heat exchanger can be made more conveniently by the NTU approach which is based on the concept of capacity ratio, effectiveness and number of transfer units. The approach facilitates the comparison between the various types of heat exchangers which may be used for a particular application.
Capacity Ratio (C):
The product m c (mass × specific heat) of a fluid flowing in a heat exchanger is repeatedly encountered and is termed as the capacity rate. It indicates the capacity of the fluid to store energy at a given rate.
Capacity rate of the hot fluid, Ch = mh ch
Capacity rate of the cold fluid, Cc = mc cc
The capacity ratio C is defined as the ratio of the minimum to maximum capacity rate. In parallel or counter flow heat exchangers, the hot or cold fluid may have the minimum value.
The temperature distribution along the exchanger length would be different depending on capacity ratio. The relative temperature change of the two fluids is inversely related to their capacity rates; the one with a smaller value of capacity rate experiencing the greater change in temperature.
When mh ch > mc cc in the counter-flow arrangement, the temperature tend to converge at the inlet. However, the temperature would be diverging when mc cc > mh ch. Further, in a parallel flow arrangement, tc2 approaches th2 for an infinitely long heat exchanger; consequently the counter-flow configuration is more effective in its operation.
Heat exchanger effectiveness (ϵ) = The effectiveness of a heat exchanger is defined as the ratio of the energy actually transferred to the maximum theoretical energy transfer.
The actual energy transfer is given by the product of the capacity rate and the temperature difference for either fluid, i.e.,
A maximum possible heat transfer (Qmax) race is achieved if a fluid undergoes temperature change equal to the maximum temperature difference available. Both for the parallel and counter- flow exchanger, the maximum temperature difference equals the inlet temperature of hot fluid minus the inlet temperature of the cold fluid, i.e.,
maximum available temperature difference = (th1 – tc1)
Further, the maximum possible heat transfer occurs when the fluid of a small heat capacity rate undergoes the maximum temperature difference available.
The maximum possible energy transfer, therefore, becomes:
Qmax = Cmin (th1 – tc1)
The effectiveness of heat exchanger is then:
Since either the hot or cold fluid may have the minimum value of capacity heat rate, there are two possible values of effectiveness
The subscript on ϵ designates the fluid which has the minimum heat capacity rate. Apparently, the effectiveness is simply a ratio of the temperature change of the fluid with the smaller heat capacity to the maximum temperature difference available in the heat exchanger.
Number of Transfer Units (NTU):
The number of transfer units (NTU) is a measure of the size of heat exchanger; it provides some indication of the size of the heat exchanger.
It is defined as,
Effectiveness for the Parallel Flow Heat Exchanger:
The heat exchange through an incremental area dA of the exchanger is
The dimensionless ratio UA/Cmin is called the number of transfer units (NTU) and the factor Cmin/Cmax represents the capacity ratio (C).
Therefore,
Where, ϵh is the effectiveness of the parallel flow exchanger with hot fluid having the minimum capacity rate. The same relationship would result when the analysis is made with the cold fluid having minimum capacity rate. The suffix h can, therefore, be dropped to give effectiveness of a parallel flow exchanger as,
Effectiveness for the Counter Flow Heat Exchanger:
The heat exchange through an incremental area dA of the exchanger is,
The integration limits for the temperature difference are from (th1 – tc2) to (th2 – tc1) and those of area from 0 to A –
Substituting these values in equation 13.41, we get,
Where, ϵc is the effectiveness of the counter flow exchanger with cold fluid having the minimum capacity rate. The same relationship would result when the analysis is made with hot fluid having minimum capacity rate. The suffix c can, therefore, be dropped to give effectiveness of a counter flow heat exchanger as,
It is to be noted that the expressions for the effectiveness contain only the overall heat transfer coefficient, area, fluid properties and flow rates.
Limiting Values of Capacity Ratio, C:
Two limiting cases of practical interest are:
(i) During the process of boiling and condensation, only a phase change takes place and one fluid remains at constant temperature throughout the exchanger.
By definition, the specific heat represents the change of enthalpy with respect to temperature, i.e., cp = dh/dt. With temperature difference dt being zero, the effective specific heat and consequently the heat capacity tends to infinity. In that case Cmax = ¥ and Cmin/Cmax = 0. The expression for effectiveness then reduces to,
ϵ = 1 – exp (- NTU) … (13.44)
both for parallel and counter flow configurations.
(ii) In a gas turbine recuperator, the exhaust gases after expansion in the turbine are used to heat the compressed air. Both the fluids have approximately equal thermal capacities and so the capacity ratio, C = Cmin/Cmax becomes very close to unity.
Then for the parallel flow configuration,
Obviously, no matter how large the exchanger be or how high be the of overflow at transfer coefficient, the maximum effectiveness for parallel flow heat exchanger is only 5%.
For counter flow arrangement, effectiveness becomes indeterminate and that necessitates fresh analysis.
The limiting value of effectiveness for counter flow exchanger is 100% and apparently counter flow units are more advantageous for gas turbine heat exchangers.
Graphs of effectiveness versus the NTU-parameters have been published for various heat exchanger configuration; the graphs for parallel and counter flow heat exchanger configurations are shown in Figs. 13.15 and 13.16 respectively. These curves indicate the relationship between
i. effectiveness ϵ
ii. Cmin/Cmax
iii. UA/Cmin = NTUmax
When any two of three parameters are known, the third can be read from these graphs. That serves to completely define the heat exchanger and its heat transfer performance.