Article pubs.acs.org/IECR
Improving Energy Efficiency and Cost-Effectiveness of Batch Distillation for Separating Wide Boiling Constituents. 1. Vapor Recompression Column Md. Malik Nawaz Khan, G. Uday Bhaskar Babu, and Amiya K. Jana* Energy and Process Engineering Laboratory, Department of Chemical Engineering, Indian Institute of Technology, Kharagpur, India 721 302 ABSTRACT: Although the direct vapor recompression column (VRC) has been known for its application in continuous distillation since the 1960s, the research on vapor recompressed batch distillation (VRBD) started a couple of years ago. In this contribution, a batch splitter assisted by a heat pump of the vapor recompression type is investigated by rigorous method. When the VRC system is employed in batch processing, a variable manipulation procedure has to be adopted, leading to the adjustment of either the vapor inflow rate to the compressor by splitting the overhead vapor or the external heat input to the reboiler. The present work aims at exploring the operational and economical feasibility of the VRBD column for the fractionation of a mixture consisting of wide boiling components. By separating this binary mixture, it is quantitatively shown how closely the reversible batch operation can be approximated by using the direct compression of overhead vapor. To further strengthen the advantages of the VRC scheme over its conventional counterpart, a variable speed VRBD structure is proposed by introducing a variable speed compressor in the VRBD configuration aiming to run the column at a controlled compression ratio.
1. INTRODUCTION Process intensification has been identified as the need to develop smaller, cleaner, and more efficient technologies.1 Because of low thermodynamic efficiency and high energy consumption, distillation is one of the natural candidates for intensification. By lowering the energy demands in the reboiler of the intensified distillation processes, such improved schemes would also have a positive and long-term effect on several factors, such as operating cost and importantly, CO2 emissions. Actually, there exists a strong correlation between the energy consumption in a distillation and the greenhouse gas emissions to the environment. Distillation is used for about 95% of all fluid separations in the chemical and allied industries; its applications ranging from the rectification of alcohol, which has been practiced since antiquity, to the separation of crude oil.2 The intensification in distillation processes is achieved by means of internal and external heat integrations. The heat integrated distillation column (HIDiC)3−12 is the most popular example of former scheme and the vapor recompression column (VRC) or heat pumping13−18 is based on the external thermal integration. Although the low compression ratio can provide benefits for the HIDiC, in general, the average vapor flow in a typical VRC column is smaller than the vapor flow at the top of the HIDiC stripper.12 Contrary to the continuous distillation process, the heat integration of batch columns has been explored very little in published literature, although the research work has started in the area of energy intensification since the 1960s. Takamatsu et al.19 first proposed a thermally coupled batch distillation configuration, in which the rectifying tower is surrounded by a jacketed reboiler. The advantage of this novel scheme over its conventional counterpart in terms of energy efficiency and cost was subsequently evaluated by Maiti et al.20 Recently, another energy efficient structure is developed for the unsteady state batch operation within the VRC framework with a unique characteristic of © 2012 American Chemical Society
operating compression ratio (CR) adaptation. This adaptive system is conceptually introduced and demonstrated with great economical potential by Jana and his co-workers21,22 under the name “variable speed” VRC. In this communication, we extend their approach to introduce the direct VRC in batch processing, and explore the economic and energetic potentials in the case of the separation of a mixture having wide boiling constituents. Additionally, here we propose for the first time the fixed speed VRC for batch distillation to present a systematic comparison with its variable speed counterpart. The heat pumping in a continuous distillation column is an economic way to conserve energy when the temperature difference between the overhead and bottom of the column is small.15 This is mainly because of the involvement of low CR in a VRC operation for separating a close-boiling mixture. To examine the suitability of VRC operation standing on the same ground, this work aims to synthesize the vapor recompressed batch distillation (VRBD) system for the separation of a feed mixture, consisting of wide boiling components. Unlike the case with continuous distillation, the VRBD structure shows promising potential in terms of thermodynamic efficiency and cost. To improve further the overall performance, in this contribution, we develop a novel variable speed VRBD for the same treatment.
2. DIRECT VAPOR RECOMPRESSION COLUMN (VRC) The conventional VRC technique depends upon the fact that in many distillation processes the heat to be removed in the trimcondenser at the top of the column is almost equal to the heat put in at the bottom by the reboiler.23 The heat input is at a higher Received: Revised: Accepted: Published: 15413
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As mentioned earlier, in the VRC system, the overhead vapor leaving the topmost tray (i.e., nTth tray) is first compressed for pressure elevation (i.e., for temperature elevation) and then the hot vapor is thermally integrated with the relatively low temperature still liquid. In the still pot, the compressed vapor releases heat through the condensation process and this latent heat leads to the reduction/avoidance of external energy input to the still. By this way, the overhead vapor can be employed as an internal source of energy after compression. To ensure the optimal use of internal heat source, there is a necessity to devise a suitable control policy for variable manipulation. In the following, we formulate a variable adjustment methodology for the VRBD column. The operation principle of the vapor recompression scheme for batch processing is schematically shown in Figure 1.
energy level, and therefore to reuse the heat rejected in the condenser, its energy level (i.e., its temperature) should be raised by some form of heat pump. Conceptually, a VRC is an externally heat-integrated scheme with vapor leaving the top of the column, compressed to a higher pressure, which upon entering the bottom reboiler starts to condense, providing the heat duty for evaporation of bottom liquid in the reboiler. This fact indicates a decrease of energy requirements from external source. Moreover, the condensation of complete or a part of overhead vapor occurred in the reboiler leads to the reduction of overhead condenser size, thereby reducing the cost of condenser (i.e., capital investment) along with its heat load (i.e., operating cost). However, the VRC system provides all these benefits at the expense of additional capital and operating costs involved in compression. We should note that the compressor requires electricity, which is several times more expensive than the thermal utility (e.g., steam and cooling water) required in running the reboiler as well as condenser. Again, the capital cost of the compressor increases nearly proportionally to the break power. All these important factors need to be taken into account when we evaluate the economical feasibility of the VRC scheme with reference to the conventional stand alone column.
3. VAPOR RECOMPRESSED BATCH DISTILLATION (VRBD): THE PROPOSED SCHEME The main drawback of the VRC scheme is that it involves the operation of heat pump over the maximum temperature difference that exists in the system. This indicates the necessity of a large compression ratio (CR) for the VRC and imposes practical limitations on the application of this technique. It is proved15,17 that the heat pumping system is energetically and economically attractive when this technique is applied over a small temperature range, particularly for separating a close-boiling mixture. However, all the points mentioned above are specific to the VRC technique involved in continuous flow distillation columns. Interestingly, in the separation of a feed mixture having wide boiling constituents, the temperature difference between the two ends of a batch column is not so large as compared to continuous distillation. The reason for this dissimilarity in nature is that throughout the entire batch distillation, the reflux drum (at the top) and the reboiler or still pot (at the bottom) never reach together at high purity levels, and therefore there exists a relatively small temperature difference between the overhead and bottom of the column. In contrast, the continuous distillation usually operates at steady state with reasonably pure (top and bottom) products, particularly for binary systems, and this leads to a large temperature difference in the case of a wide boiling mixture separation. This fact motivates us to explore the operational and economic feasibility of the energy integrated VRC technology for batch distillation with the treatment of a mixture consisting of wide boiling components. As indicated above, compared to a continuous column, a batch rectifier under the VRC scheme requires lower compression ratio when the boiling point temperatures of the components to be separated are far apart. This, in turn, decreases the electricity cost as well as the capital investment for the compression unit. To investigate the overall performance improvement achieved by the VRBD column over the regular batch distillation, we aim to conduct a comparative study between them on the basis of identical process dynamics. For this purpose, we will attempt to keep the input conditions (e.g., amount of feed charged, feed composition, and reboiler duty) and the output specifications (e.g., product purity and flow rate) the same.
Figure 1. Schematic representation of the proposed VRBD scheme.
Control Variable Manipulation. We assume that the condensing vapor starts changing its phase in the still pot when there is a thermal driving force of at least 20 °C; that is, ΔTC ≥ 20 °C. This driving force (ΔTC) corresponds to the temperature difference between the compressed overhead vapor (TnTC) and the reboiler liquid (TB). It is a fact that the ΔTC (= TnTC − TB) plays a vital role in determining the compression ratio from CR =
PnTC PnT
⎛ Tn C ⎞ μ /(μ− 1) = ⎜⎜ T ⎟⎟ ⎝ TnT ⎠
(1)
Here, the stream pressures PnT and PnTC are specific to TnT (overhead vapor temperature) and TnTC, respectively, and μ is the polytropic coefficient that is temperature dependent.21 In a batch distillation, both the TnT and TB vary because of the transient behavior of batch processing. It is proposed that the VRBD operates with a fixed-speed vapor compressor. Now to meet the operating criterion concerning ΔTC ≥ 20 °C, we use eq 1 to fix the CR value that corresponds to the maximum ΔTT (= TB − TnT). Interestingly, this is the largest CR value attained at a particular time instant within the stipulated batch time. 15414
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Figure 2. Proposed energy utilization policy with variable manipulation mechanism.
required to manipulate in the first scenario (when QCV > QR) to avoid the entry of excess heat (i.e., QCV − QR) to the reboiler that might provide deteriorated dynamic performance with longer batch time. On the other hand, in the second scenario (when QCV < QR), it is suggested to control the inflow rate of external heating medium to supply the makeup heat to the still pot.
The optimal use of internal energy source is the main target of any heat integration approach. In the heat pumping arrangement, this can be accomplished by extracting the latent heat of overhead vapor and utilizing it in bottom liquid reboiling. Anyway, to operate the vapor recompressed batch column at a fixed CR and constant reboiler heat duty (QR), we propose an effective energy utilization policy. In this manipulation mechanism, we explore the existence of two scenarios, as detailed below. Scenario 1 (when the heat released by the compressed vapor (QCV) is more than the reboiler energy demand (i.e., QCV > QR)): In this scenario, the vapor stream leaving the topmost tray (VnT) can provide more than the required heat. For the typical batch operation at a fixed QR, therefore, a part of the overhead vapor (VnTi), which corresponds to an amount of latent heat of QCV − QR, should be directed to the overhead condenser. The remainder of vapor (VnTC) that can exactly meet the reboiler energy demand can be utilized as an internal heat source. Based on this concept, we have VnTi = VnT − VnTC
4. VARIABLE SPEED VRBD COLUMN: THE IMPROVED SCHEME For boosting further the overall performance, we extend the VRBD technique to the variable speed vapor recompression column. In this adaptive VRBD system, in addition to the fixed QR, another criterion concerning ΔTC of exactly 20 °C is also taken into consideration. The main purpose to set the ΔTC at 20 °C is to run the column at a controlled compression ratio so that the unnecessary overheating of overhead vapor can be avoided before thermally coupling it with the reboiler liquid. To meet this second objective, we need to adapt the CR at every time step based on the extended form of eq 1 as
(2)
CR =
where VnTC =
QR λ n TC
PnTC PnT
⎛ Tn C ⎞ μ /(μ − 1) ⎛ ΔT + T ⎞ μ /(μ − 1) B⎟ = ⎜⎜ T ⎟⎟ = ⎜⎜ C ⎟ T T ⎝ nT ⎠ ⎝ ⎠ nT
(4)
So, for implementing the proposed variable speed VRBD system, we should additionally manipulate the CR with controlling the QCV and QME as presented earlier under the direct VRBD column. It is fairly true to say that in this variable speed configuration, along with the operating CR, either the overhead vapor flow to the compressor or the external heat input to the reboiler are required to adjust simultaneously. Actually, the VRBD operates with constant QR and CR, whereas the variable speed scheme aims to meet the criteria concerning fixed QR and ΔTC.
(3)
Here λ represents the latent heat. By this way, the splitting of the overhead vapor is made to manipulate the VnTC for controlling the QR at its desired value. Scenario 2 (when QCV < QR): In this class of operation, the reboiler demands more heat than the heat available from the internal source. To run the VRBD at the same dynamical performance with the regular batch column, an external heating medium, therefore, is required for supplying the makeup energy, QME (= QR − QCV) to the still pot. Although this scenario additionally involves an external heat source, ideally it does not require any overhead condenser. In this control variable manipulation procedure proposed for the VRBD and briefly summarized in Figure 2, the VnTC is
5. AN ILLUSTRATIVE EXAMPLE: BINARY MIXTURE WITH WIDE BOILING CONSTITUENTS 5.1. Base Case. To demonstrate the features of the proposed VRBD configuration, a simple example, separating a mixture of methanol and water, is considered. We chose this system because it is a typical separation of compounds with wide boiling points, 15415
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liquid holdup, fast energy dynamics, atmospheric pressure at the top with a stage pressure drop of 0.3 kPa, nonlinear Francis-weir formula for tray hydraulics, Wilson model to describe thermodynamic properties, and constant liquid holdup in the reflux drum. The modeling equations24 for the conventional batch distillation are presented in the Appendix for a typical nth tray. The operating parameters and column specifications, summarized in Table 1, are used in the computer simulation. The operation of a regular batch distillation is usually divided into two phases, namely the startup and production phases. In the former phase, the column runs in complete reflux mode and the later phase involves the batch processing under partial reflux conditions. During total reflux operation, the concentration of the lightest component increases on the upper trays of the column and the concentrations of the other components increase in the still pot. When the distillate composition of low boiler reaches its desired purity, then the reflux ratio is set to a prespecified value and at the same time, the distillate product is taken out. In the present study, the column runs in startup mode until it reaches the steady state. Subsequently, the production phase starts and the product withdrawal continues till the average
where the vapor recompressed continuous distillation is not an economically attractive option.15 Therefore, we are interested to assess the economic feasibility of the heat pumping operation in batch mode for this commonly used system. The representative column has a total of seven stages; the bottommost tray is Stage 1 and the topmost tray is Stage 7. At the beginning, the binary feed mixture is charged in the reboiler and condenser, and on the trays. The mathematical model is formulated on the basis of the following assumptions: perfect mixing and equilibrium on all stages, negligible tray vapor holdup, variable Table 1. System and Column Specifications system total feed charge, kmol feed composition (startup), mol fraction tray holdup in each tray (startup), kmol reflux drum holdup, kmol tray efficiency, % heat input to the still pot, kJ/min distillate composition (steady state), mol fraction distillate rate (fixed), mol/min
methanol/water 90.0 0.45/0.55 0.225 2.7 70 2000.0 0.993 8.0
Figure 3. Methanol composition profile of the regular batch distillation. 15416
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Figure 4. Regular batch column profile in terms of ΔTT.
Figure 5. Variation of ΔTC in the VRBD at a fixed CR of 2.16.
and CR. The value of QR (2000 kJ/min) is adopted from the base case. It is now necessary to determine what is the maximum value of ΔTT, based on which the operating CR can be found out. For this, we produce Figure 4 that demonstrates the temperature profile throughout the batch operation. In this figure, we identify the tth time step (i.e., the end point) at which the ΔTT is highest. It is interesting to note that the maximum ΔTT obtained (25.8 °C) is lower than the difference in normal boiling point between the two wide boiling components, methanol and water (i.e., 35 °C). Furthermore, looking at the complete profile in Figure 4, it becomes obvious that the ΔTT on an average is reasonably low. This fact would indicate the involvement of lower CR in VRC for batch processing compared to continuous distillation operation. This advantage is further strengthened in the case of variable speed VRBD arrangement. As stated, the proposed thermal integration scheme should perform the batch processing in order to meet the two criteria concerning ΔTC and QR. Accordingly, we first set the CR value using eq 1 at 2.16 that can provide the ΔTC of 20 °C at tth time instant. By performing a simulation experiment at this fixed operating CR value, we obtain Figure 5 that clearly shows the VRBD operation with ensuring ΔTC ≥ 20 °C throughout the entire batch run.
distillate composition remains above the specified purity. For the example system, as shown in Figure 3, the column attains steady state taking nearly 615 min (startup period) with a methanol composition of 99.3% in the top product. As stated previously, for a meaningful comparison between all the concerned batch configurations, along with the input conditions, we attempt to keep the output specifications (i.e., product amount and average composition) identical. Accordingly, the production phase runs with a constant distillate rate of 8 mol/min (i.e., 14.71% of steady state VnT). By performing the simulation experiment, the production period of 4570 min is obtained on the basis of the average methanol top composition fixed at 98%. It can be seen from the simulation results that at the end of batch operation (i.e., at the end of 5185 min (batch time)), the water purity in the reboiler reaches at 94.3%. However, at the same time, the average composition of the whole column contents should be lower than 94.3%. To purify the second product (i.e., water) further, therefore, it is suggested to run the batch distillation with continuing the liquid distillate collection as a slop-cut beyond 5185 min until the column reaches the water purity at a specified level. 5.2. Vapor Recompressed Batch Distillation. In this work, we propose that the VRBD is to be operated at a fixed QR 15417
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Figure 6. QCV profile at a fixed CR of 2.16.
Figure 7. Compressed overhead vapor profile in all three column configurations.
with the same dynamical performance. By analyzing the example binary system, we investigate that the proposed VRBD operated at a fixed CR manipulates the QE, not the V7C. 5.3. Variable Speed VRBD Arrangement. Recall that the VRBD structure operates with fixed QR and CR, whereas the QR and ΔTC are held constant in case of the variable speed VRBD system. In this improved heat integration scheme, along with the QE, additionally the CR needs to be adjusted for the example system. Actually, the variable speed configuration considers the ΔTC of exactly 20 °C, instead of ΔTC ≥ 20 °C used in the VRBD system, in order to avoid the compression operation at a highest CR with unnecessary overheating of condensing vapor. By conducting simulation experiments, we produce the manipulated variables profiles for the variable speed VRBD column. Figures 8 and 9 demonstrate the adjustment of QE and CR, respectively. Analyzing the results, one may readily find the coincidence in QE between the VRBD and variable speed scheme occurred at the end point. This happens because of the operation
Now, to operate the vapor recompressed batch column at a constant reboiler duty of 2000 kJ/min, the variable manipulation methodology as detailed earlier needs to be followed. Running the nonintegrated batch column at the CR of 2.16, we get Figure 6, in which one may readily find that the QCV is lower than QR in all times. However, although QCV < QR, the difference between them is significantly low with a maximum deviation of about 98 kJ/min that is merely 4.9% of QR. It quantitatively shows how closely the reversible operation can be approximated by using the direct compression of overhead vapor. It is evident that the representative system involves only Scenario 2 because of QCV < QR throughout the operation. To ensure an exact reboiler heat consumption of 2000 kJ/min for the complete batch time, the VRBD operation utilizes the whole overhead vapor as an internal heat source in addition to the external heat of QE (= QR − QCV) to supply the makeup energy to the still pot. The column profile in terms of V7C (=V7) is illustrated in Figure 7. Note that the variation of V7C is identical for all three configurations because of their operations carried out 15418
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Figure 8. Manipulation of external heat input to the still pot of variable speed VRBD column.
Figure 9. Manipulation of compression ratio (CR) in the variable speed VRBD column.
In this equation, the QComp is in hp, the pressure (inlet pressure, P7 and outlet pressure, P7C) is in lbf/ft2, and the overhead vapor flow rate (V7) is in ft3/min. For the heat integrated column, we adopt f = 3 and
of both thermally coupled columns at time =5185 min with the same ΔTC of 20 °C and QR of 2000 kJ/min. 5.4. Thermodynamic Efficiency. As mentioned previously, the batch operation comprises startup and production periods. The total heat consumed by a batch rectifier (Qcons) is calculated by summing up the heat duties involved in both startup and production periods. Again, the heat load in each of these two phases (Q) combines the heat supplied by an external medium (QE) and the compressor load (QComp) based on the following correlation:
Q = Q E + fQ Comp
QE = 0 Q E = Q ME
⎧ QE = QR ⎪ ⎨ ⎪Q ⎩ Comp = 0
where Q Comp
(for Scenario 2)
As suggested by Iwakabe et al.,9 it is logical to adopt f = 3 by assuming that 3 kW of thermal energy is needed to produce 1 kW of electrical power. On the other hand, in the case of the regular batch column,
(5)
μ − 1/ μ ⎡⎛ ⎤ P7C ⎞ μ ⎢ = 3.03 × 10 − 1⎥ V7P7 ⎜ ⎟ ⎢⎝ P7 ⎠ ⎥ μ−1 ⎣ ⎦
(for Scenario 1)
−5
On the basis of this concept, we obtain the QCons for VRBD and regular batch column as 2.24 × 106 kJ and 10.37 × 106 kJ,
(6) 15419
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respectively. Accordingly, with reference to the regular column, the heat integrated scheme secures a significant energy savings of 78.4%. It should be highlighted that the proposed heat integrated VRBD shows a good approximation of process reversibility. We observe that about 97% (total QE = 3.3 × 105 kJ) of reboiler heat demand can be met simply by the internal energy source, leaving an insignificant amount of external heat input to the still. Likewise, by simulating the representative batch rectifier under variable speed vapor recompression mode, we obtain 83.54% reduction in energy consumption, indicating nearly 5% more energy savings by the variable speed scheme over the VRBD system. This additional energy savings is achieved by the variable speed column because of the operation of the compressor at optimal CR values. 5.5. Economics. The economic evaluation of each configuration is conducted in terms of the total annual cost (TAC) by combining the annual capital and operational costs. By adopting the basic column dimensions (e.g., diameter, height, efficiency, tray design, and spacing) of the regular batch column (base case), and assuming the same for both VRC schemes, we estimate the costs using the correlations given by Douglas25 and documented in Table 2. Additionally, the capital investment
Table 3. Comparative Capital ($) and Operational Costs ($/yr) regular batch distillation column shell column tray compressor reboiler condenser total steam cooling water electricity total TACa TAC savings, % a
Capital Costs 31223.68 31223.68 1127.23 1127.23 0.0 18662.60 13826.31 13826.31 16515.21 0.0 62692.43 64839.82 Operational Costs 5571.46 176.79 583.73 0.0 0.0 1667.89 6155.19 1844.68 27052.67 23457.95 13.29
variable speed VRBD 31223.68 1127.23 18313.86 13826.31 0.0 64491.08 142.48 0.0 1236.29 1378.77 22875.80 15.44
For a payback period of 3 years.
savings in TAC. As far as the variable speed VRBD scheme is concerned, more energy (83.54%) and TAC (15.44%) savings have been achieved compared to the VRBD column. This improvement is achieved mainly because of the overhead vapor compression at varying speeds, leading to lower compressor duty and electricity costs.
Table 2. Cost Estimating Formula and Parameter Value column shell (MOC, SS) ⎛ M&S ⎞ 1.066 0.802 ⎟101.9D installed cost ($) = ⎜ Lc (c in + cmc p) c ⎝ 280 ⎠
6. CONCLUSIONS This study mainly concerns exploring the economic feasibility of heat pump in batch distillation to reduce energy use. Although the direct VRC scheme is not an attractive option for a wide boiling mixture separation in a continuous distillation because of the existence of large temperature difference between the two ends of the column, this problem is not so prominent in the case of the proposed VRBD system. Unlike the continuous column, the batch processing never achieves high purity at a time in both the top reflux and the reboiler, thereby reducing the CR and improving the thermodynamic efficiency. In vapor recompressed batch distillation, either the vapor inflow rate to the compressor or the heat input to the still pot is required to control aiming to run the column at a constant QR. Actually, the VRBD configuration operates with a constant reboiler duty and compression ratio. To improve further the economical potentials, we introduce the variable speed VRBD system, in which, along with the QR, the ΔTC is kept constant. To operate the batch rectifier with a variable speed compressor, along with either the VnTC or QE, additionally the CR needs to be manipulated. Both the proposed VRC schemes are demonstrated with the methanol−water system. It is observed that the VRBD and its variable speed counterpart secure an energy savings of about 78.4% and 83.54%, respectively, and a TAC savings of 13.29% and 15.44%, respectively. When the CR is allowed to vary throughout the batch processing, better energy and economic performance can be obtained. The vapor recompression solution is highly attractive compared to the conventional distillation. However, the economic benefits from heat pump system come at the expense of difficulties in operation, posing a major hurdle in their widespread usage.
where Dc is the column diameter, Lc is the column height, M&S = 950, and the coefficients cin = 2.18, cm = 3.67, and cp = 1.0. column tray (type, bubble-cap; MOC, SS) ⎛ M&S ⎞ 1.55 ⎟4.7D installed cost ($) = ⎜ c Lc(cs + c t + cm) ⎝ 280 ⎠ where the coefficients cs =1, ct = 1.8, and cm = 1.7. heat exchanger (MOC, SS) ⎛ M&S ⎞ 0.65 ⎟101.3A installed cost ($) = ⎜ (c in + cm(ct + c p)) ⎝ 280 ⎠ where the coefficients cin = 2.29, cm = 3.75, ct = 1.35, and cp = 0. compressor
installed cost ($) =
VRBD
⎛ M&S ⎞ 0.82 ⎜ ⎟517.5 (bhp) (2.11 + Fd) ⎝ 280 ⎠
where Fd = 1.0. This expression is valid in the range of 30 < bhp
