The Numerical Psychrometric Analysis

Transcription

1 he Numerical sychrometric Analysis by Jorge R. López Busó, MSME, E Introduction he sychrometric Analysis is the base of any HVAC system design. Nowadays, the psychrometric analysis is mainly done by means of he sychrometric Chart. he sychrometric Chart contains in a graphical form an approximation of the properties of moist air. hose approximations are currently available in the form of equations. With the very well established computer technology and in favor to the fame of he sychrometric Chart, means to solve the equations using numerical algorithms are not described readily elsewhere. his lack of welldocumented algorithms is also a reason why the psychrometric analysis has not crossed over into the computer technology, representative of the almost past three decades. here is no doubt that the simplicity of the sychrometric Chart is a welcomed featured. But as HVAC technology has evolved and variable volume systems are getting more accepted, new means to perform the calculations required for such systems are necessary. his article presents the background necessary for the numerical solution of the equations that approximate moist air properties. In doing so I am presenting a series of simple algorithms that makes use of existing equations. Moreover, I have used those algorithms to solve a problem that has been for many years debated by mechanical and energy engineers. roperties Moist Air he following section is a brief presentation of the equations used to approximate the properties of moist air. hese equations can be obtain from ASHRAE Fundamentals 1997 and reference 2. he Enthalpy and the Specific Volume equations will not be used in this article but they are shown for completeness. a)! Vapor ressure(v) - he vapor pressure of water has been approximated using the saturated vapor pressure of water at a given temperature. he saturated vapor pressure of water at temperatures above 32 F is given in sia by: ( ) V t R = e & C $ % t C 9 + C10' tr + C11' tr + C12' tr 3 R + C13' Ln # ( t )! R " Eq Jorge R. Lopez 1

2 he constants C8 through C13 are as follows: C8 = C9 = C10 = C11 = C12 = E-09 C13 = t R = tf b)! artial Vapor ressure(v) - he partial vapor pressure of water can be approximated using Carrier's partial vapor pressure equation. he partial vapor pressure is given in sia using the following equation: where: B w = V( ) V (, ) = = Dry Bulb emperature ( F) = Wet Bulb emperature ( F) = Barometric ressure (sia) W " ( " )!( " ) B W 2831" 143.! (Saturated Air Vapor ressure) Eq. 2 c)! Humidity Ratio(W) - he humidity ratio measures the amount of moisture in the air. he humidity ratio in Lb. water /Lb. dry air can be calculated as follows: where: = V V ( ) W V, = " Eq. 3 B! V ( ) (Moist Air Vapor ressure), d)! Relative Humidity(!) - he relative humidity in percentage is calculated using the following equation: V "(, ) =! 100 Eq. 4 2 VS

3 where: = V VS ( ) (Dry Air Vapor ressure) e)! Enthalpy (H) - he enthalpy of air in Btu/Hr-Lb. dry air is given by: H ( W ) = 0.24! + ( * ) W, Eq. 5! f) Specific Volume (SV) - he specific volume of air in ft 3 /Lb. dry air is given by: SV ( W ) = ! ( ) ( ! W ),! Eq. 6 B Numerical Algorithms Of interest for the sychrometric Analysis is the determination of the dry bulb temperature if the relative humidity and the humidity ratio are known. he relative humidity and the humidity ratio are indirect functions of vapor pressure. Because there is no simple way to obtain an equation for temperature as a function of vapor pressure (ie. (V) as opposed to Eq. 1, V()), an iterative technique has to be used to solve the equations. he technique used in this article is to re-group the equations so that an error function can be generated to iterate on using the Bisection Method. Because the humidity ratio is known, equation 3 can be used to find the moist air vapor pressure ( V ). hus one obtains that: V W! B = Eq W Also, because relative humidity is known, the dry air vapor pressure VS can be calculated using Eq. 4 and the result obtained above. he following is obtained: V =!100 Eq. 7 " VS 3

4 We can now form our error function for dry bulb temperature based on dry air vapor pressure as follows: W * B E VS ( ) = VS ( ) "! 100 Eq. 8!# ( W ) he Bisection Method iteration is very simple. It starts with a possible range for the variable we want to iterate on, in this case. An acceptable range for the solutions we are looking is 32 F to 100 F. he upper limit can be increased, but in order to decrease the lower limit, a different equation has to be used for the function V(). On every iteration, the Bisection Method evaluates the error function on the lower limit and the middle of the range. hen it compares the signs of the values obtained and selects on which side of the range the solution is. his is continued until some convergence criteria is minimized or certain number of iterations are done. he following listing is a cookbook recipe to he Bisection Method for this specific case. 1.! Set iteration number to zero. 2.! Assign low limit to 1 and the high limit to 2. 3.! Calculates the mid range value ( 3 ) as follows: = Eq ! Evaluate Error Function (Eq. 8) at 1 and 3 to obtain E VS1 ( 1 ) and E VS3 ( 3 ). 5.! If E VS1 ( 1 )"E VS3 ( 3 ) is less than zero then 2 = 3 else 1 = 3. 6.! Calculate relative error criteria: REC = E VS3 ( 3 )/ 3. 7.! Increase number of iterations by one. 8.! If REC is greater than a preset tolerance (10-7 ) and number of iterations is less than maximum (200) then goto step 2. 9.! If the number of iterations is less than maximum, a solution was found and is equal to 3. else iterations did not converged to a solution 4

5 Another useful function for the sychrometric Analysis is the determination of wet bulb temperature based on dry bulb temperature and relative humidity. Because the Bisection Method has already been described above, I will only show the Error Function. ( ) ( ) " = # ( ) Eq VS E V! V, Uses of the Numerical Methods In a standard cooling cycle a mix of outside air and return air enters the coil where it is cooled and dehumidified. Depending on the room sensible heat ratio (RSHR) and coil conditions, the air will require reheat in order to satisfy room latent loads. o remove the required moisture from the entering air (to satisfy room conditions), the air has to be cooled down below the sensible load requirements of the room. his happens because a coil can only remove moisture when its apparatus temperature is below the dew point of the air. he effect of this is that the air leaving the coil is at a relative humidity that is close to the 100% saturation curve. A common assumption is that the air leaves the coil at 95% relative humidity. A plot of the system air conditions in the sychrometric Chart will look like Figure 1. Outside Air Coil Air Mix Room Reheat Figure 1. Standard Cooling Cycle It can be clearly seen that the coil leaving conditions will depend on the room sensible heat ratio. Because of limitations in the cooling equipment performance and economic factors, there is going to be a range of room sensible heat ratios where reheat will be required. his has a lot of importance on energy conservation because reheat is usually seen as a waste of energy. 5

6 Many debates have been developed around the need for reheat in HVAC systems. As we are going to see, the use of numerical methods can help discern when reheat is required and when it is not. Based on the standard cooling cycle presented on Figure 1 and previous discussion, the numerical methods presented in this article can be used to solve this problem. Within this context, the following variables govern the standard cooling cycle.! W RSHR! R, R CL, CL W R, R CL, CL = Room Dry Bulb emperature ( F) = Room Wet Bulb emperature ( F) = Room Relative Humidity (%) = Room Humidity Ratio (Lb. water/lb. da.) = Room Sensible Heat Ratio, = Coil Leaving Dry Bulb emperature ( F) = Coil LeavingWet Bulb emperature ( F) = Coil Leaving Relative Humidity (%) = QS RSHR = Q + Q Coil Leaving Humidity Ratio (Lb. water/lb. da.) S L Of these, the coil leaving dry bulb temperature, the coil leaving wet bulb temperature and the coil leaving humidity ratio are unknown. he coil leaving relative humidity will be fixed at 95%. he room conditions will be set at 75 F dry bulb temperature and 50% relative humidity. Equation 10 is used with he Bisection Method to find room wet bulb temperature. Equation 3 is used to find room humidity ratio. But then, how does one finds the temperature at which the room process line (RSHR line) intersects the 95% relative humidity curve? If you calculate the required air flow rate using the sensible load equation (Eq. 11) and the latent load equation (Eq. 12), both should be equal in order to maintain the room sensible heat ratio. By matching these two equations and using the definition of room sensible heat ratio, an error function can be generated to calculate coil leaving dry bulb temperature. Q s = 1. 1" CFM "! Eq. 11 Q L = 4760 " CFM "! W Eq. 12 6

7 he error function to calculate the coil leaving dry bulb temperature is: ( ) RSHR" ( W! W (,# )) R CL CL, CL E CL CL, = R,!! CL, 1.1(1! RSHR) Eq. 13 It can be noticed that this error function is independent of the air flow rate and the actual room loads. It is only dependent on three factors, namely coil leaving relative humidity, room humidity ratio and room sensible heat ratio. his allows for a general formulation. able 1 values were calculated making use of the error function presented in equation 13. As it can be seen, beyond sensible heat ratios of 0.64, it is impossible to obtain a solution if there is no reheat. More over, the normal range of operation of chilled water coil leaving conditions is in the 52 F to 56 F. In this range if the room sensible heat ratio drops below 0.82, the system will require the use of reheat. With DX coils, you can go down to 45 F and consequently you should not use reheat unless the room sensible heat ratio drops below hose ranges are standard practices that allow for the economic factor to stay within acceptable ranges. Keeping in mind these limits, one can only question where would sensible heat ratios of less than 0.82 be found? Example roblem Lets take as an example the case of a Casino and assume the following things: 1.! Floor area of 6500 square feet. 2.! eople density of 15 sq. ft. per person (433 persons total). 3.! eople loads as follows: Sensible 275Btu/Hr-person Latent 275Btu/Hr-person 4.! Light density of 1 watt per square feet (6500 watts total). 5.! Equipment load of 25,000 watts. 6.! No solar load (a night calculation). 7.! Wall transmission is negligible. 8.! Any heat storage effects are neglected. 9.! Required room conditions of 75, 50%RH. 10.!Coil leaving relative humidity is 95%. 7

8 able 1. Coil Leaving Conditions for Various Room Sensible Heat Ratios Room Sensible Heat Ratio Dry Bulb emperature ( F) Wet Bulb emperature ( F) Relative Humidity (%) Humidity Ratio (Grains/Lb.da)

9 When the load calculations are performed it can be readily seen that the sensible heat ratio is close to Let's also assume that the cooling equipment we are dealing with is capable of 48 F coil leaving dry bulb temperature. With those restrictions if you look in able 1, you will notice that reheat is going to be required (ie. For RSHR=0.65, CL, =39.28 F) Moreover, if equation 13 is modified to allow the specification of reheat, one would see that a reheat between 5 F and 6 F is required. It is worth to note that the required CFM for this case can be calculated with equation 11. It would not be a surprise that the AHU to be selected based on the CFM for this case would require a double coil. his is another constrain that the engineer will have to work with the equipment manufacturer. In the event that no reheat would be provided, the resulting room conditions would yield a high room relative humidity. In some cases it can be as high as nearing the saturation curve. he importance of this assessment is that you can run into condensation problems if off-load conditions are over-looked and reheat is not taken into consideration. Because reheat is determined by the lowest room sensible heat ratio, it is important to look for off-load requirements in any system. In helping solve this problem I developed Figure 2. Figure 2 shows what happens when reheat is added to equation 13. he figure provides coil leaving dry bulb temperature for various amount of reheat with varying room sensible heat ratios. With this figure one can readily calculate the coil leaving conditions and adjust any system for minimum reheat. An ideal use of this would be in control schemes. By means of a reset on coil leaving temperature calculated using this algorithm, humidity can be controlled and reheat can be minimized. Conclusions he use of computer algorithms to analyze the psychrometric behavior of HVAC systems is a powerful tool. By properly re-grouping moist air property equations one can solve any cooling cycle. Of interest to the engineering community, the methods presented were applied to the standard cooling cycle. It was shown that reheat is a required need for low room sensible heat ratio conditions. It was implied that such conditions might be found in off-load conditions and that as such it should be investigated how off-load room sensible heat ratio varies. he algorithms presented in this article could be applied in control systems to minimize reheat. More investigation is needed along these lines to improve current design procedures. 9

10 Coil%Leaving%Dry%Bulb%emperature%(95%RH) Dry%Bulb%emperature%( F) Reheat/=/0.0/ F Reheat/=/1.0 F Reheat/=/2.0 F Reheat/=/3.0 F Reheat/=/4.0 F Reheat/=/5.0 F Reheat/=/6.0 F Reheat/=/7.0 F Reheat/=/8.0 F Reheat/=/9.0 F Reheat/=/10.0 F Room%Sensible%Heat%Ratio

11 References 1.! American Society of Heating, Refrigerating and Air Conditioning Engineers, ASHRAE Handbook: Fundamentals 1997, Atlanta, USA, ! Clifford, George, Modern Heating, Ventilating and Air Conditioning, rentice Hall, New Jersey, USA, 1990.