Spot-on: Safe Fuel/Air Compression

Transcription

1 Spot-on: Safe Fuel/Air Compreion Problem preented by Robert Hart and Kevin Hughe Veeder-Root Participant: Jeffrey Bank Joeph Fehribach Alitair Fitt John Ockendon Colin Pleae Don Schwendeman Burt Tilley Suzanne Weeke Report prepared by B. Tilley, received 22 January 2004 Abtract The report begin by decribing a fundamental mathematical model for the Veeder-Root problem. It i baed on a crude adiabatic model with no heat loe. Thi model predict that the preure in the torage tank after n cycle i given by (32) and Figure 3. The only mechanim for the temperature rie i the work done againt compreion during each cycle. The effect of heat loe to the exterior are then modeled in increaing order of complexity. It i unclear which i the mot important heat lo mechanim but all the one conidered how a relatively mall perturbation to the adiabatic reult. The mot dangerou part of the equipment i the valve, whoe face i repeatedly expoed to the hottet vapor in each cycle. An etimate ha been given of the temperature of thi hotpot in Figure 0 and. The poibility of uing a 2-tage compreor to operate at lower temperature wa dicued. Thi poe an intereting and quite new mathematical challenge to be addreed in the future. Introduction The emiion of fuel vapor into the atmophere from underground torage tank at filling tation i a common occurrence in many part the world. The condition of the vapor in the tank vary ignificantly over a 24 hour period uch that evaporation and exce air ingetion during the refueling proce can caue tank over preurization and ubequent emiion. At other time during a 24 hour cycle, preure can fall below atmopheric preure. The tate of California ha recognized thi emiion problem and ha enacted regulation to addre it. Due to thee low-emiion environmental requirement in California, olution mut be implemented that do not entail releae of thee vapor into the atmophere. One olution require that the vapor fill a balloon during the appropriate time. However, the ize of the balloon at typical inflation rate require a ignificant amount of phyical pace (approximately liter), which may not necearily be available at filling tation in urban area. Veeder-Root ha a patent pending for a ytem to compre the vapor that are releaed to a 0: ratio, tore thi compreed vapor in a mall torage tank, and then return the vapor to the original underground fuel tank when the condition 29

2 Figure : Schematic diagram of the ytem to be invetigated are thermodynamically appropriate (ee Figure for the chematic repreentation of thi ytem). The limitation of the compreor, however, i that the compreion phae mut take place below the ignition temperature of the vapor. For a 0: compreion ratio, however, the adiabatic temperature rie of a vapor would be above the ignition temperature. Mathematical modeling i neceary here to etimate the performance of the compreor, and to ugget path in deign for improvement. Thi report tart with a mathematical formulation of an ideal compreor, and ue the anticipated geometry of the compreor to tate a implified et of partial differential equation. The adiabatic cae i then conidered, auming that the temporary torage tank i kept at a contant temperature. Next, the heat tranfer from the compreion chamber through the compreor wall i incorporated into the model. Finally, we conider the cae near the valve wall, which i ubject to the maximum temperature rie over the etimated 0,000 cycle that will be neceary for the proce to occur. We find that for adiabatic condition, there i a hot pot cloe to the wall where the vapor temperature can exceed the wall temperature. Latly, we dicu the implication of our analyi, and it limitation. 30

3 2 Problem Formulation We tart by dicuing the general problem for the idealized cae of a ingle pump with imple inlet and outlet valve. Thee valve are aumed to be completely open whenever the flow i in the correct direction while they cloe whenever the reulting preure drop would make the flow in the wrong direction. The inlet i connected to an ource of ambient ga mixture and the outlet to a torage tank. The main model to be conidered i et out in the equation (2), (3), (4) and (5) but we give a more detailed derivation for the intereted reader. Conider the ga mixture in the domain x p(t ) < x < L, where the piton i located at x = x p(t ) (ee Figure 2. The piton ha a troke length l < L, and thi length need to be determined baed on the overall compreor capabilitie. Surrounding thi ga in a cylindrical container are the compreor wall, found in the domain a < r < a + b, 0 < x < L The equation of motion for the ga are given by conervation of ma, conervation of momentum, conervation of energy, and an equation of tate ρ t + (ρ u ) = 0 () ρ u t + u u } + p = 0 (2) ρ c vo Tt T } + p ( u ) = k o 2 T (3) p = Rρ T, (4) where ρ i the denity of the mixture, u = (u, v ) i the ga velocity (here axial and radial component are taken and we aume no circumferential flow), p i the ga preure, c vo i the pecific heat of the ga at contant volume, T i the ga temperature, k o i the thermal conductivity of the ga, and R = c po c vo i the univeral ga contant. Here we have conidered an invicid fluid and in practice the flow i anticipated to be turbulent. If vicou effect are to be included care hould be taken to include both hear and bulk vicoity effect. Thi ytem i coupled to the energy equation in the cylindrical hell: ρ w c pw T w t = k w 2 T w, (5) where ρ w i the denity of the material of the wall, c pw i the pecific heat at contant preure in the wall, T w i the wall temperature, and k w i the thermal conductivity of the wall. There are four tage in each cycle which correpond to change in the boundary data. In the top-left diagram in Figure 2 we ee a repreentation of tage. During thi tage, the ga i compreed, and both valve remain cloed. When the preure of the ga in the compreor increae to be equal to the preure in the torage tank the exhaut valve open and the vapor flow into the torage tank. Thi i the tart of tage 2, hown in the top-right diagram of Figure 2, which begin when the piton i a ditance l + F from the compreor wall, where F need to be determined Stage 2 continue until the piton ha reached x = l and then tage 3 begin a the piton revere direction. During tage 3 the preure i below the preure in the torage tank and both valve remain cloed (ee the bottom-left figure in Figure 2). When the preure ha decreae to a value equal to the ambient preure of the inlet ga the inlet valve open and tage 3 end. During 3

4 Figure 2: Repreentation of the four tage of the compreion cycle. Top-left diagram: Stage, the vapor at ambient condition initially i compreed to the torage-tank preure; Top-right diagram: Stage 2, the exhaut valve now open, and the vapor i dicharged into the torage tank; Bottom-left diagram, Stage 3, the piton revere direction, the exhaut valve cloe, and the remaining vapor expand and cool until the ambient preure i reached; Stage 4, the inlet valve open, bringing freh ga at ambient preure into the tank to be compreed in the next cycle. 32

5 Material Quantity Unit Value ρ o kg/m 3.29 Air (at 300 K) c vo J/(kg K) 000 k o W /(m K) κ o m 2 / ρ w kg/m Cat Aluminum (9.5% Al) c pw J/(kg K) k w W/(m K) 50 κ w m 2 / h W/(m 2 K) - 0 Deign Quantity Variable Unit Value Compreor Length L m 0. Piton Stroke l m Piton Radiu a m Compreion Caing Thickne b m Piton Cro-ectional Area A p m Storage Tank Volume V m Piton Stroke Frequency f Characteritic Piton Velocity U p m/ 0. Table : Dimenional material and deign propertie. tage 4 the inlet valve i open and the ga mixture, at ambient condition, enter the tank. Stage 4, a hown in the bottom-right diagram of Figure 2, tart when the piton i a ditance G from the compreor wall, where G need to be determined. Stage 4 continue until the piton return to the original poition at the tart of tage, at which point the cycle repeat. Mathematically, thee tage can be, in an ideal cae, decribed a: Stage : Compreion. t o < t < t x = x p(t ) : u = d dt x p(t ), v = 0 T x = 0 x = L : u = 0 v = 0 T x = 0 r = 0 : r T r = 0 r = a : v r = 0 T = T w k o r T r = k w r T w r v = 0 r = a + b : k w T wr = h (T w T o ), 33

6 Stage 2: Iobaric Exhaut. t < t < t 2 Stage 3: Expanion t 2 < t < t 3 x = x p(t ) : u = d dt x p(t ), v = 0 T x = 0 x = L : v = 0 T x = 0 r = 0 : r T r = 0 r = a : v r = 0 T = T w k o r T r = k w r T w r v = 0 r = a + b : k w T wr = h (T w T o ), x = x p(t ) : u = d dt x p(t ), v = 0 T x = 0 x = L : u = 0 v = 0 T x = 0 r = 0 : r T r = 0 r = a : Stage 4: Iobaric Intake. t 3 < t < t 4 v r = 0 T = T w k o r T r = k w r T w r v = 0 r = a + b : k w T wr = h (T w T o ), x = x p(t ) : u = d dt x p(t ), v = 0 T x = 0 x = L : v = 0 T x = 0 r = 0 : r T r = 0 r = a : v r = 0 T = T w k o r T r = k w r T wr v = 0 r = a + b : k w T wr = h (T w T o ), 34

7 where h i the heat tranfer coefficient between the outer compreor wall and the environment. Note here we have taken the compreor inner urface of the piton (x = x p(t ) and the valve region (x = L) to be thermally inulated but thee could be readily taken to be imilar to the cylindrical urface. The critical problem to conider i the heat tranport from the compreing vapor to it urrounding. With thi in mind, we hall focu on the heat tranport apect of thi ytem, and not concern ourelve with the fluid-mechanical detail of the ytem. We perform a caling analyi to how that thi approach i jutified. We cale x on L, r on a, t on /f (where f i the frequency of the piton ocillation), u on U p = L f, and ga propertie on their ambient value ρ o, p o, T o uch that p o = Rρ o T o. We hall drop the upercripted notation to indicate that, from here on, nondimenional quantitie are being conidered. From thi nondimenionalization, the reulting equation are in the ga x p (t) < x <, 0 < r <, t > 0: ρ ρ o U 2 p p o ρ o U 2 p p o T t + ut x + L a vt r ρ t + (ρ u) x + L a r (rρ v) r = 0 (6) ρ u t + uu x + L } a vu r + p x = 0 (7) ρ v t + uv x + L } a vv r + L a p r = 0 (8) } + R ( p u x + L ) c vo a r (rv) r = κ } o L 2 L U p a 2 r (rt r) r + T xx (9) p = ρ T, (0) where κ o = k o /(ρ o c vo ) i the thermal diffuivity of the vapor. In the cylindrical hell, we have that T wt = κ } w L 2 L U p a 2 r (rt wr) r + T wxx, () where κ w = k w /(ρ w c pw ) i the thermal diffuivity of the wall material. The boundary condition then become (only Stage i hown below): x = x p (t) : u = x p(t), v = 0 T x = 0 x = : u = 0 v = 0 T x = 0 r = 0 : r T r = 0, v r = 0 r = : T = T w k o r T r = r T wr k w v = 0 r = + b a : T w r = h a (T w ) k w 35

8 Note that for the ytem at hand, Up 2 ρ o /p o 0 8, L/a 0, and κ/(lu p ) 0 3. Further, for cat aluminum, we find that κ w /(LU p ) 0 3. Thu, to leading order both momentum equation give u that p x = p r = 0, or that p = p(t). The leading-order conervation of ma equation give rρv = A, and, ince v = 0 on r =, it follow that A = 0. In addition ince ρ 0, by the equation of tate, we have that v = 0. The reulting equation then become: ρ t + (ρ u) x = 0 (2) p = p(t) (3) ρ T t + ut x } + (γ ) p u x = λ o r (rt r) r + ɛ 2 T xx } (4) p = ρ T, (5) where ɛ = a/l i the apect ratio of the compreor, and λ o = (Lκ o )/(a 2 U p ) i the ratio of the rate of heat propagation to advection. Thi, along with the heat equation in the wall: T wt = λ w r (rt wr) r + ɛ 2 T wxx }, (6) where λ w = (Lκ w )/(a 2 U p ) i the ratio of heat tranport through the cylinder to advection. The boundary condition for thi ytem are (again, only Stage i hown here): x = x p (t) : u = x p(t), T x = 0 (7) x = : u = 0 (8) T x = 0 (9) r = 0 : r T r = 0 (20) r = : T = T w (2) K r T r = r T wr (22) r = + d : T wr = B (T w ) (23) where K = k o /k w 0 4 i the ratio of thermal conductivitie of the wall material to the vapor, B = h a/k w 0 3 i the Biot number, and d = b/a i the relative thickne of the apparatu caing to it radiu. 36

9 Quantity Name Variable Typical Value Mach Number Squared M 2 = ( ρou 2 p µu p L p o p o ) Vicou Term 0 0 Specific Heat Ratio γ.4 Vapor Thermal Diffuion λ o 0.3 Apect Ratio ɛ 0. Volume Ratio V = (LA p /V ) Scaled Wall Thickne d = b/a 0.2 Wall Thermal Diffuion λ w 0.08 Thermal Conductivity Ratio K Biot Number (heat tranfer) B Reult Table 2: Dimenionle quantitie and their typical value We now conidered the problem that ha been poed in different ituation. The aim here to give relatively imple reult that can be interpreted phyically and alo to look at everal different apect of the behavior. 3. Perfectly Inulated Sytem The implet cae of the ytem (2) - (6) ubject to the boundary condition (7)-(23), during all four tage, i to aume that there i no heat tranport into the wall of the compreor or to outide. Thi cae correpond to taking λ o = λ w = B = K = 0, o there i no heat flux through the ide wall and the wall have no heat capacity. We can then aume that the temperature i independent of both x and r reulting in the following et of equation: ρ t + (ρ u) x = 0 p = p(t) ρt t + (γ ) pu x = 0 p = ρt. Note that for γ =, we have the iothermal cae and we hall include thi cae in the reult below. We identify the boundary condition that occur at each tage: Stage : t o < t < t : u(x p (t), t) = x p(t), u(, t) = 0, (24) 37

10 Stage 2: t < t < t 2 : u(x p (t), t) = x p(t), (25) Stage 3: t 2 < t < t 3 : u(x p (t), t) = x p(t), u(, t) = 0, (26) Stage 4: P t 3 < t < t 4 : u(x p (t), t) = x p(t), (27) Note that the time t, t 3 are determined implicitly by p(t ) = P, p(t 3 ) =, where i the preure within the torage tank during cycle n. We are going to find how the preure in the tank increae from one cycle to the next through the compreion proce. During tage, u(x p (t), t) = x p(t), u(, t) = 0. We can olve for the velocity u from the conervation of ma relation u = ρ t ρ (x ), and the velocity condition at the piton reult in ρ(t) = x p (t). Uing thi information and the equation of tate, the energy equation become or that ρt t (γ ) ρ t T = 0, T (t) = ρ γ. Thi with the equation of tate reult in the tandard adiabatic olution Stage end when p(t ) = P p(t) = ρ γ.. Hence during all of tage 2 we will have p(t) = P. u(x, t) = x p(t) identically, and the denity and temperature will not vary. For implicity we define the following quantitie: For Stage 2 we note that at time t = t, p(t ) = P F = l x p (t ) (28) T = T (t ) (29) ρ = ρ(t ). (30) During Stage 3, the ga expand, and the olution to the preure, denity and temperature are given by ρ(t) = ρ ( l) x p (t), T (t) = ρ γ, p(t) = ρ γ. (3) 38

11 Latly, during Stage 4, the input valve open, and ambient material enter the ytem, giving that ρ =, p =, T =. We now conider how the preure in the torage tank increae. We note that during tage 2 the ma added to the torage tank i given by M = ρ F V, where V = (LA p /V ) i the ratio of the volume in the compreor to that within the torage tank. With thi added ma, and the aumption that the torage tank i large enough to dipere any additional heat o that the ga in it i under iothermal condition, we can calculate the torage tank preure a: P (n+) = P + V ρ F. Further we note that the that the exhaut valve open when p(t ) = P. Thi give a relation between F, ρ, and P and at t = t : or : p(t ) = P = (ρ(t )) γ = ( ρ ) γ ρ(t ) = ρ = ( F = ρ = ( P P = ) (/γ), x p (t ) l F ) /γ ( l). Hence, our firt-order difference ytem for the torage-tank preure i given by P (n+) = P + V ( l) [ ] P /γ }. (32) In Figure 3 we how the trajectory of the torage tank preure P over a typical filling proce plotted againt the cycle n for variou γ from iothermal (γ = ) to the adiabatic cae for air γ =.4. In the actual application, where the ga contain a mixture of primarily n-butane (C 4 H 0 ), Pentane (C 5 H 2 ) and Hexane (C 6 H 4 ) at a volume fraction of about 20%, a weighted average for the pecific heat ratio give γ.3. In the lower figure, we plot the temperature T at the beginning of the exhaut tage for the n th cycle. Notice that for the mixture in quetion that the final temperature at a 0-fold compreion i approximately 530 K. Thi final temperature i near the auto-ignition point of the mixture, which i not deirable for table operating condition. 3.2 Effect of the Compreor Wall The previou analyi aumed that there wa no heat tranfer to or from the ga mixture. However, the time cale on which the torage tank fill (approximately 0,000 ) i comparable to the typical time cale of heat tranfer through the compreor wall and out 39

12 P γ =.0 γ =. γ =.2 2 γ =.3 γ = Number of Cycle.5 2 x γ =.4 T γ =.2 γ = γ =. 300 γ = Number of Cycle x 0 4 Figure 3: Preure and Temperature rie for the adiabatic cae, independent of wall temperature, for γ =, γ =., γ =.2, γ =.3, and γ =.4 with l = (0.) γ and V = For a 20% mixture of n-butane, Pentane, and Hexane with dry air (γ.3), the final temperature reache approximately 530 K. 40

13 to the environment. Since thee effect occur on the ame time cale, their competition i important to undertand. In the following ection we look at thi apect on the cale of the whole compreor. We will aume that the patial dependence of ρ, u x, p and T i not ignificant in the majority of the compreion chamber o that averaged dependent variable depending only on time (except of u) can be ued. We will however till account for the heat flux through the chamber wall. We tart with equation (2)-(5), multiply each equation by r and integrate over 0 < r < to arrive at the following ytem of equation, where ρ, u, p and T are aume to be averaged value in r: ρ t + (ρ u) x = 0 (33) p x = p(t) (34) ρ T t + ut x } + (γ ) p u x = 2λ o Q (35) p = ρ T, (36) where Q = rt r r= i a meaure of the amount of heat tranfer that occur between the vapor in the compreor and the compreor wall. We further aume that, although the temperature may vary over each cycle, that the average temperature of the cylinder wall, T w = +d rt w dr, varie lowly in time. We can ee thi by multiplying the heat equation within the cylinder wall by r and integrating over thi domain to find that T w t = 2 λ w 2d + d 2 B[T w(, t) ] K Q(t)}. (37) Since B, K = O(V ) 0 3, the average wall temperature within the cylinder change lowly over time t. For the following, we make the approximation that T w (, t) T w, and if we integrate (37) over one cycle in time, then we can write T (n+) w T w = 2 λ w 2d + d 2 B [ T w ] + KQ }, (38) where we have approximated the average temperature, T w, by T w and the average value of the heat flux, Q, by Q. In the following, we need an additional relation to determine the heat flux into the compreor wall. We conider only two cae here. The firt i to aume that heat tranfer to the wall i relatively rapid o that the average wall temperature, T, i ame a the average vapor temperature, T, over the cycle. Thi give a contant heat flux into the wall over each cycle and we call thi the quai-teady Q problem and i the implet ytem to analyze. A econd approach i to aume that the heat flow i related to the temperature of the vapor and the wall through a Newton law of cooling approach Q = α(t T w ), where α i the heat tranfer parameter from the wall to the vapor, and would need to be determined by experiment. Again, the behavior in the ga need to be determined during each cycle, and the reulting analyi then coupled to the wall temperature through (38). Notice that T w in thi analyi i quai-teady by (37). In both approache, a ytem of dicrete equation i derived that decribe the wall and vapor temperature at each cycle n. 4

14 3.2. Quai-teady Q The following analyi i the eaiet cae to olve (33)-(36) with (38) and give ome inight into the behavior of the ytem. Note that with the aumption that we have made, we can olve for T in equation (35) directly with T = T i ( ρ ρ i ) γ + 2 λ o ρ γ t t i Q ρ γ d t, (39) where the i ubcript correpond to the value at the beginning at that tage. In the following iobaric condition (preure i contant) are aumed whenever a valve i opened. Further, we aume that the piton peed i contant (either ±), and hence we can relate piton poition directly with time. Thi aumption i convenient for the averaging proce. During tage, both valve are cloed, and the denity and temperature are given by ρ = ρ o x p (t), t o < t < t, and the form of the temperature in thi region i given by (39). Thi formulation i valid up until t = t, when the preure p(t ) = P. At thi time, x p (t ) = l F, which give a relation between F and P P = [p o + 2 λ ] [ o Q γ + : l + F ] γ 2 λ o Q [ ] l + F. (40) γ + During tage 2, t < t < t 2 = t o + /2, p(t) = P, which give u an equation for the denity ρ: ( ) γ ρ t = ρ T + 2 λ o Q ρ γ ρ P t ρ γ d t = ρt. (4) Differentiating (4) once with repect to t give an ordinary differential equation to olve for ρ: ρ (t) = 2 λ o Q ρ, γ P reult in the following equation for the denity and temperature during tage 2: } } ρ(t) = ρ exp 2λ o Q (t t γp ), T (t) = P 2λ o Q exp (t t ρ γp ). (42) Note that thi reult in the velocity u not being uniform over x,. u = x p(t) + 2λ o Q [x x p (t)]. (43) γp 42

15 Hence, the total ma M that i put into the torage tank during thi compreion cycle i given by M = = t2 t t2 t ρ(t)u(, t)dt [ρx p (t)] ρ (t)dt [ = ρ F + (l ) exp } ]} 2 λ oq F γp, where ρ = ρ. Given that the time increae in the torage tank preure i given by P = V M, we have that [ } ]} P (n+) = P + V ρ F + (l ) exp 2 λ oq F. (44) γp Finally, the denity and temperature at the end of thi tage i given by } } ρ 2 = ρ(t 2 ) = ρ exp 2 λ o Q F 2 λ o Q, T γp 2 = T (t 2 ) = T exp F γp. Stage 3, t 2 < t < t 3, i where ome care mut be hown. Since the thermodynamic proce i not reverible here, we need to quantify how far the piton move toward x = 0 while both valve remain cloed. We denote the location x p (t 3 ) = G, where G need to be determined by the condition p(t 3 ) = p o. The analyi i imilar to Stage, and we find the equation for G to be given by [ ] γ ( l p o = P + 2 λ o Q ) } G γ+ ( l) γ+ G (G ) γ, (45) γ + and the reulting temperature field become T (t) = T 2 ( ρ ρ 2 ) γ + 2 λ o Q ρ γ t t 2 ρ γ d t. (46) Finally, during tage 4, p(t) = p o, and the denity i governed by thi iobaric condition, a in tage 2, and we find for the denity and the temperature that ρ = ρ 3 exp 2 λ } o Q (t t 3 ) γp o, T (t) = p } o 2 λo Q exp (t t 3 ) ρ 3 γp o. (47) Latly, we need to integrate thi temperature T (t) over the period t o < t < t 4 = t o +. 43

16 Performing thi integration,uing the relation above, give the reult that over one period [ T w = l + F ] 3 ( l + F ) 2 γ + 2λ o Q } + 2 γ + 2λ o Q 3 p o (γ + ) ( ) 2 γ P ( [ l + F ) exp 2λ o Q p o [ (G ( l) ) ] γ 2 γ ( l) 2 γ [ 2λ o Q 2 γ 3( l)(γ + )P γp o G 2( l)λ o Q [ exp 2 γ 2λ o Q γp F } + 2λ o Q (γ + ) P ( G ) 3 ( l) 3} } [ l + F ] P ] ] + + p o exp 2λo Q } [ ( G ) ] l + F ] exp γp o p o 2λ o Q γp F } + } 2λ o Q F. (48) γp Thu, the ytem of equation in T w (n+), P (n+), F, G, Q which we need to olve i given by (48) and: T (n+) w T w = 2 λ w [ B T 2d + d 2 w ] + KQ } (49) [ ]} P (n+) = P + V ρ F + (l ) P = p o = P ρ (n+) = exp } 2 λ oq F γp (50) [p o + 2 λ ] [ ] o Q γ 2 λ o Q [ ] l + F (5) γ + l + F γ + [ ] γ ( l + 2 λ o Q ) } G γ+ ( l) γ+ G (G ) γ (52) γ + ρ ( l) exp 2 λ [ o Q F + ( ]} G ) (53) ( l + F (n+) )G γ p o A ample calculation of thi ytem i hown in Figure 4 for larger Biot number (B = ). The torage-tank preure, the wall temperature, and the relative piton location when the exhaut and inlet valve open, F and G repectively, are hown with olid curve while the dahed curve correpond to the adiabatic cae (no heat flow from the vapor to the wall). Notice that the ytem ceae to function beyond around 3500 cycle, at which point G = and the intake valve no longer open. Thi reult in a terminal preure of about 6 atm. However, the effect of the heat tranfer i apparent in the wall temperature, which never exceed 320 K during thi entire proce. Note that the value of the Biot number i exceeding large in thi example. When the Biot number i maller, however, we find that the compreor fail due to overheating. In Figure 5 we ee the cae for B = Notice that the toragetank preure peak around 3 atm, the wall temperature tart to exceed the adiabatic Note that the olution to thi ytem wa found by converting it to a differential-algebraic ytem, and uing the Matlab olver ode5 on thi effective ytem P 44

17 P Number of cycle T w F0.4 G Number of cycle Number of cycle Figure 4: Storage-tank preure, wall temperature, and piton poition F and G for a quai-teady heat flux Q (olid line), plotted againt the adiabatic reult (dotted line) with B = Notice that the compreor continue operation longer than the adiabatic cae, but eventually return to the original temperature. At the end of the proce, the input valve top opening. temperature rie, and with F = 0, the exhaut valve ceae to open. Given the autoignition point of the vapor mixture, the mixture would combut within 3000 cycle Newton Law: Q = α(t T w ) When the heat flux Q i aumed to depend on the vapor temperature, then the olution of the temperature, in term of the denity, i given by ( ) γ ρ t T (t) = T i exp ρ i t i } 2λ o α t ρ d t + ρ γ t i t 2λ o αt w ρ γ exp t } 2λ o α ρ dτ d t. Notice that the evolution of the vapor temperature i determined by the evolution of the denity. From thi baic olution for the vapor temperature, and auming that x p(t) = ±, we can follow the ame proce in the quai-teady cae above to find an effective relation between the torage-tank preure P, wall temperature T w, the piton location F, G and the denity ρ a T (n+) w T w = 2 λ [ ] w 2d + d 2 B T w + KQ } (54) 45

18 P T w Number of cycle Number of cycle F Number of cycle G Number of cycle Figure 5: Storage-tank preure, wall temperature, and piton poition F and G for a quai-teady heat flux Q (olid line), plotted againt the adiabatic reult (dotted line) with B = Notice that the temperature begin to increae dratically, and that the compreor ceae operation ince the exhaut valve doe not open. 46

19 P (n+) = P + V ρ F (l ) [ ( γp ρ ) ( exp })]} 2 λ oα T γp w F (55) P = ( p o ( l + F ) exp λo α [ l + F ] )} 2 + γ ρ o 2λ o αt w ( λo α [ l + F ] )} 2 ( t) 2 [ l + T w l F ] γ ( t) γ exp 0 ρ o d t (56) ρ 2 = γp ρ o ρ =, T l + F = P (57) ρ + (ρ γp )e 2λoα γp F, T 2 = P ρ 2, (58) p o = ( ) γ ( l P 2λo α l + G exp ρ 2 ( l) 2 ( ) γ ρ2 ( l) G ( l) 2λ o T w G 0 [ ] [ G ( l) G + ( l)] )} + ( ρ2 ( l) G u exp 2λo α ( ( l)u + 2 )}) ρ 2 ( l) u2 du(59) Q = α 2 T w ρ 4 = γp o + (ρ 3 γp o ) exp ρ 3 = ρ 2( l), T G 3 = p o (60) ρ } 3 2λαT w G, T 4 = p o (6) γp o ρ 4 [ (T o + T )(l F ) + (T + T 2 )F + (T 2 + T 3 )[G ( l)] + (T 3 + T 4 )( G )] } (62) Notice that we ued a crude average of the vapor temperature in the definition of the heat-tranfer variable Q. In Figure 6 we how the torage-tank preure, the wall temperature, and the value of F and G for the adiabatic cae and the cae for α = 0.25, 0.25, 0.5. Interetingly, the behavior of thee cae appear independent of the heat tranfer coefficient α. In Figure 7, we ee the effect of a mall Biot number (B = ). Notice that the compreor will fail by not allowing new vapor to enter the chamber. However, Figure 8 how the vapor temperature of both large and mall Biot number with α = Notice that the mall Biot-number cae reult in the vapor temperature exceeding the auto-ignition point quite dramatically, while with ufficient heat tranfer, the vapor temperature remain below the adiabatic rie temperature Varying heat flux (Maximum temperature) A a final model of the overall effect of the compreor wall we conider a cae where we can find an upper bound of the poible ga temperature. On the long time cale of 47

20 α = 0.25 α = 0 α = 0.25 α = P T w Number of cycle F Number of cycle Number of cycle G Number of cycle Figure 6: Storage-tank preure, wall temperature, and piton poition F and G auming that heat flux i proportional to the ga-wall temperature difference for α = 0.25 (olid line), α = 0.25 (dotted line), α = 0.5 (dahed-dotted line), and the adiabatic cae (dahed line) with γ =.3 and B = Notice that the behavior of thi model reemble the adiabatic reult reaonably well. 48

21 T w 400 P Number of cycle Number of cycle F 0.5 G Number of cycle Number of cycle Figure 7: Storage-tank preure, wall temperature, and piton poition F and G auming that heat flux i proportional to the ga-wall temperature difference for α = 0.25 (olid line) and the adiabatic cae (dahed line) with γ =.3 and B = The wall temperature i again tarting to increae markedly, while the pump will fail due to the inlet valve ceaing to open G =. 49

22 Vapor Temperature B= Adiabatic 350 B= Number of cycle Figure 8: Vapor temperature a a function of cycle for α = 0.25 and mall Biot number (B = ), large Biot number (B = ) and the adiabatic cae. Notice that initially the vapor temperature mut exceed the adiabatic temperature in order to reach the torage-tank preure. After thi hort tranient, the long-time heat tranport of the compreor wall dictate the functionality of the apparatu. thouand of pump cycle, the temperature of the pump wall near the inlet/outlet end will increae due to the heat of compreion of the ga; once thi occur, the hot wall heat the ga prior to compreion creating a feedback loop. We eek to find the reulting temperature after thi very long time. A a wort cae, let u aume that the heat tranfer i uch that the wall near the inlet/outlet x = reach the maximum temperature experienced by the ga, T M. We will further aume that during the cycle the wall temperature doe not vary ignificantly and that heat i conducted into the ga from the hot wall. We will now etimate (or more pecifically find an upper bound for) the maximum temperature, T M. Late in the proce, the region l < x < along the cylindrical portion of the wall and the end-wall will attain thi temperature T M. Thi localized region, auming that the heat capacity of the compreor wall i much larger than that of the ga, provide a ource of heat during the intake cycle. Auming that the piton move at a contant peed x (t) = ±, then the energy equation for the ga i given by ρt t = α (T M T ), (63) where ( ) 2 α = a + hc, L( l) c vo ρ o where h C i the heat tranfer coefficient between the ga and the compreor wall, which need to be determined experimentally. In the following, we aume initially that the denity of the ga remain contant during the intake proce. We hall compare thi reult to that for a contant preure intake later in thi ubection. 50

23 The intake phae lat a duration G, where G = ( l) p /γ and hence the temperature at the end of thi phae T 4 i given by (T M T 4 ) = (T M )exp α( G)}. in the adiabatic cae, The intake phae i followed by the compreion phae. If the compreion i adiabatic from p = to p = p and that the temperature rie back to T M, one find that γ γ T 4 = T M p. Combining the previou two equation, one obtain an explicit formula for T M (uing the adiabatic form of G): [ γ γ T M = ( p )exp α( ( l)p /γ ) }]. (64) A imilar analyi, auming that the intake proce i iobaric (ρ = /T ) yield the following expreion between T M and T 4 : T M = T M (T M T 4 ) e αt M ( G) (T M T 4 )e αt M ( G) + T 4. (65) Applying an adiabatic compreion phae (3.2.3) reult in the relation γ γ ( p )e αt M ( G) T M = T M γ γ ( p γ )e αt M ( G) γ + p, (66) which implifie to [ γ ] T M = γ (p ) exp α T M ( G)} +. In figure 9, we how in the reult for α = 0.25, 0.25 and 0.5, l = 0.9, and uing the adiabatic reult for the ditance G at which the intake valve open. The upper figure correpond to the iochoric cae, while the lower figure correpond to the iobaric cae. Notice that in the iochoric regime, the bound on the maximum temperature (auming that the compreor wall reache a teady-tate maximum temperature) i larger than that of the adiabatic temperature (olid curve). However, in the iobaric cae, the maximum poible temperature i actually cooler than the iochoric cae. Thi can be undertood eaily, ince the denity increae during thi tage. The equation of tate then ugget that, ince preure i contant, the temperature decreae during thi tage. 3.3 Thermal Boundary Layer near Valve Wall At thi tage, we have conidered imple ordinary differential equation model that do not take into account any gradient in pace of either denity, temperature, or preure of the vapor. However, during the operation of the compreor, we expect that the temperature 5

24 T M P T M P Figure 9: Comparion of the maximum temperature calculation for the iochoric intake cae (upper figure) and the iobaric intake cae (lower figure). In both cae, the olid curve correpond to the adiabatic limit (α = 0), while the dahed curve, dotted curve, and dahed-dotted curve correpond to α = 0.25, 0.25, and 0.5 repectively. Notice that in the iochoric cae, if the maximum temperature doe exit, it may be higher than the adiabatic temperature. However, in the iobaric cae, the maximum temperature, if it exit, can be lower than the adiabatic cae. 52

25 of the valve wall, which i expoed mot often to the hot vapor, will become ignificant. Hot pot within the vapor can reult during the compreion tage, ince the ga within the chamber i heated, and there will be ome heat tranfer from the valve wall to the ga. The combination of thee effect reult in a localized region where the ga become heated above the wall and adiabatic temperature. A proper compreor deign will need to take thi effect into account. To undertand thi phenomenon we conider a thin boundary layer of ga adjacent to the end wall, auming that there i behavior in only one direction. In addition we take the wall temperature to be contant at T, which i near the maximum adiabatic rie temperature and expected to be in the range.5 < T < 2 (again, caled on 300 K). with boundary condition ρ(t t + ut x ) + (γ )pu x = λ o ɛ 2 x κ(ρ) T } x (67) ρ t + ρu} x = 0 (68) p = ρt (69) p = p(t), (70) T (x, 0) =, ρ(x, 0) =, u(0, t) = 0, T (0, t) = T, where each quantity ha been caled baed on the adiabatic condition prior to the operation of the compreor, and ɛ = a/l i the apect ratio of the compreor, which i mall. To perform the boundary-layer analyi, we aume that which then give u that: ξ = x ɛ λ o, u = ɛ λ o ū, ρ(t t + ūt ξ ) + (γ )pū ξ = ξ ρ t + ξ κ(ρ) T } ξ (7) ρū} = 0 (72) ( ) γ p = ρt = (73) t p = p(t). (74) Note that p, ρ and T have not been recaled, but u ha. We require at the boundary condition ξ = 0 that ū = 0, and that T = T. A ξ, we need that ( ) γ T T adiabatic =, ū ξ t t. T (0, t) = T, T T adiabatic a ξ. 53

26 From thi anatz, we can ue the fact that p i given to reduce the equation. If we multiply the conervation of ma equation (72) by T and add the reult to the heat equation (7), the we find that p (t) + γp(t)ū ξ = (k(t )T ξ ) ξ. We can integrate thi equation once in ξ to find that ū = γp(t) ξp (t) + k(t )T ξ q(t)}, where q(t) i an integration contant that need to be determined from matching to the outer problem. However, we do not necearily have enough information to perform thi matching. However, if we apply the boundary condition at ξ = 0, we find that q(t) = k[t (0, t)] T ξ (0, t), or that q(t) i a meaure of the heat flux leaving the wall. Note that if we were to aume the wall wa inulated, rather than at contant temperature T, then we would know q(t) = 0, however, here we mut find thi quantity. To find the equation for the temperature (or denity), we can ue thee expreion for ū and ū ξ in the conervation in ma equation (72). From thi, we find that (uing the equation of tate (73)), p (t)t p(t)t t T 2 + p ( T ūξ + ū p(t) ) T T 2 ξ = 0, and after multiplying by T 2, and uing the appropriate expreion for ū, ū ξ, we find that ] γp(t)t t = (γ )p (t)t (x, t)+ [T k(t )T ξ } ξ + ξp (t)t ξ k(t ) T 2ξ + q(t) T ξ ubject to the boundary condition T (0, t) = T, T T adiabatic, a ξ., ρ = p(t) T (ξ, t), Note that in practice the thermal conductivity of the ga mixture varie with temperature and for the calculation hown here we have taken k(t ) + 2T. Thi i appropriate auming that the dominant thermal behavior i due to the nitrogen in the vapor. In Figure 0 we how the evolution of one particular cae, with T =.5 (we plot the reult in the phyical temperature, meaured in Kelvin) and equally paced interval in time from t = 0 to t = 0.5. Notice that the development of the hot pot near the hot wall a the bulk temperature approache it maximum adiabatic value. In Figure, we note that the maximum temperature value, compared to the wall temperature, increae a the wall temperature increae, from about 7% for T =.5 to nearly 2% for T = 2. 54

27 500 t = T (in K) t = ξ Figure 0: Temperature profile T a a function of the inner variable ξ during the compreion tage. Notice that near the end of the compreion cycle that a region near the wall i heated above the adiabatic temperature (T max T )/T T Figure : Percentage temperature rie within the thermal boundary layer a a function of wall temperature T. Notice that a the wall i heated that the relative temperature rie aturate near 2%. 55

28 4 Dicuion In thi report, we have outlined ytematically a one-dimenional approach addreing the feaibility of the compreor needed for the Veeder-Root emiion application. From thi analyi, we have found that there i a good poibility, provided that a ufficient amount of heat tranport from the apparatu to the environment i attained, for a 6: compreion ratio. Thi reult depend however, on the aumption that have been made in thi analyi. Thee include the neglect of i) radial effect within the compreion chamber, ii) fluid-dynamical effect near the entrance or exit valve, iii) heat tranport through the piton, iv) no material change that occur between the piton and the chamber wall (i.e. chafing) and v) any axial dependence of the heat tranport. However, the model preented in thi work i amenable to including thee effect if further analyi i warranted. Hence, the reult from thi report hould be ued a an initial guide for deign: it i poible that vapor temperature may indeed be larger than reported here when additional phyical effect are included. Further, we have found that there i a good likelihood of a hot-pot, whoe temperature i approximately 0% higher than the hottet part of the apparatu. Thi reult limit the potential of what the compreor could realitically handle. In concluion, the requirement for a compreor hown in Figure hould be attainable either through a ingle tage compreor (with a larger torage tank than firt uggeted), or by uing a econd compreion tage. The analyi included in thi work i pertinent to a multi-tage compreor olution to the problem. 56