Heat and mass transfer scale-up issues during freeze-drying, III: Control and characterization of dryer differences via operational qualification tests

The objective of this research was to estimate differences in heat and mass transfer between freeze dryers due to inherent design characteristics using data obtained from sublimation tests. This study also aimed to provide guidelines for convenient scale-up of the freeze-drying process. Data obtained from sublimation tests performed on laboratory-scale, pilot, and production freeze dryers were used to evaluate various heat and mass transfer parameters: nonuniformity in shelf surface temperatures, resistance of pipe, refrigeration system, and condenser. Emissivity measurements of relevant surfaces such as the chamber wall and the freeze dryer door were taken to evaluate the impact of atypical radiation heat transfer during scale-up. “Hot” and “cold” spots were identified on the shelf surface of different freeze dryers, and the impact of variation in shelf surface temperatures on the primary drying time and the product temperature during primary drying was studied. Calculations performed using emissivity measurements on different freeze dryers suggest that a front vial in the laboratory lyophilizer received 1.8 times more heat than a front vial in a manufacturing freeze dryer operating at a shelf temperature of −25°C and a chamber pressure of 150 mTorr during primary drying. Therefore, front vials in the laboratory are much more atypical than front vials in manufacturing. Steady-state heat and mass transfer equations were used to study a combination of different scaleup issues pertinent during lyophilization cycles commonly used for the freeze-drying of pharmaceuticals.     

Evaluation of manometric temperature measurement (MTM), a process analytical technology tool in freeze drying, part III: Heat and mass transfer measurement

This article evaluates the procedures for determining the vial heat transfer coefficient and the extent of primary drying through manometric temperature measurement (MTM). The vial heat transfer coefficients (K_v) were calculated from the MTM-determined temperature and resistance and compared with K_v values determined by a gravimetric method. The differences between the MTM vial heat transfer coefficients and the gravimetric values are large at low shelf temperature but smaller when higher shelf temperatures were used. The differences also became smaller at higher chamber pressure and smaller when higher resistance materials were being freeze-dried. In all cases, using thermal shields greatly improved the accuracy of the MTM K_v measurement. With use of thermal shields, the thickness of the frozen layer calculated from MTM is in good agreement with values obtained gravimetrically. The heat transfer coefficient “error” is largely a direct result of the error in the dry layer resistance (ie, MTM-determined resistance is too low). This problem can be minimized if thermal shields are used for freeze-drying. With suitable use of thermal shields, accurate K_v values are obtained by MTM; thus allowing accurate calculations of heat and mass flow rates. The extent of primary drying can be monitored by real-time calculation of the amount of remaining ice using MTM data, thus providing a process analytical tool that greatly improves the freeze-drying process design and control.     

Spatial Variation of Pressure in the Lyophilization Product Chamber Part 2: Experimental Measurements and Implications for Scale-up and Batch Uniformity

Product temperature during the primary drying step of freeze-drying is controlled by a set point chamber pressure and shelf temperature. However, recent computational modeling suggests a possible variation in local chamber pressure. The current work presents an experimental verification of the local chamber pressure gradients in a lab-scale freeze-dryer. Pressure differences between the center and the edges of a lab-scale freeze-dryer shelf were measured as a function of sublimation flux and clearance between the sublimation front and the shelf above. A modest 3-mTorr difference in pressure was observed as the sublimation flux was doubled from 0.5 to 1.0 kg·h^?1·m^?2 at a clearance of 2.6 cm. Further, at a constant sublimation flux of 1.0 kg·h^?1·m^?2, an 8-fold increase in the pressure drop was observed across the shelf as the clearance was decreased from 4 to 1.6 cm. Scale-up of the pressure variation from lab- to a manufacturing-scale freeze-dryer predicted an increased uniformity in drying rates across the batch for two frequently used pharmaceutical excipients (mannitol and sucrose at 5% w/w). However, at an atypical condition of shelf temperature of +10°C and chamber pressure of 50 mTorr, the product temperature in the center vials was calculated to be a degree higher than the edge vial for a low resistance product, thus reversing the typical edge and center vial behavior. Thus, the effect of local pressure variation is more significant at the manufacturing-scale than at a lab-scale and accounting for the contribution of variations in the local chamber pressures can improve success in scale-up.     

Impact of Natural Variations in Freeze-Drying Parameters on Product Temperature History: Application of Quasi Steady-State Heat and Mass Transfer and Simple Statistics

Inter- and intra-batch variability in heat and mass transfer during the drying phase of lyophilization is well recognized. Heat transfer variability between individual vials in the same batch arise from both different positions in the vial array and from variations in the bottom contour of the vials, both effects contributing roughly equally to variations in the effective heat transfer coefficient of the vials, K_v. Both effects can be measured in the laboratory, and variations in average K_v values as a function of vial position in the array for lab and production can be calculated by use of the simple steady-state heat and mass transfer theory. Typically, in the laboratory dryer, vials on the edge of the array, “edge vials,” run 2–4°C warmer than “center vials,” but differences between laboratory and manufacturing temperatures are modest. The variability in mass transfer can be assigned to major variations in ice nucleation temperature (both intra-batch and inter-batch), including major differences between laboratory and manufacturing. The net effect of all random variations, for each class of vial, can be evaluated by a simple statistical model-propagation of error, which then allows prediction of the distribution in product temperatures and drying times, and therefore prediction of percent of vials dry and percent of vials collapsed and proximity to the edge of failure for a given process. Good agreement between theoretical and experimentally determined maximum temperatures in primary drying and percent collapsed product demonstrates the calculations have useful accuracy.     

Evaluation of manometric temperature measurement, a process analytical technology tool for freeze-drying: Part II measurement of dry-layer resistance

The purpose of this work was to study the factors that may cause systematic errors in the manometric temperature measurement (MTM) procedure used to determine product dry-layer resistance to vapor flow. Product temperature and dry-layer resistance were obtained using MTM software installed on a laboratory freeze-dryer. The MTM resistance values were compared with the resistance values obtained using the “vial method.” The product dry-layer resistances obtained by MTM, assuming fixed temperature difference (ΔT; 2°C), were lower than the actual values, especially when the product temperatures and sublimation rates were low, but with ΔT determined from the pressure rise data, more accurate results were obtained. MTM resistance values were generally lower than the values obtained with the vial method, particularly whenever freeze-drying was conducted under conditions that produced large variations in product temperature (ie, low shelf temperature, low chamber pressure, and without thermal shields). In an experiment designed to magnify temperature heterogeneity, MTM resistance values were much lower than the simple average of the product resistances. However, in experiments where product temperatures were homogenous, good agreement between MTM and “vial-method” resistances was obtained. The reason for the low MTM resistance problem is the fast vapor pressure rise from a few “warm” edge vials or vials with low resistance. With proper use of thermal shields, and the evaluation of ΔT from the data, MTM resistance data are accurate. Thus, the MTM method for determining dry-layer resistance is a useful tool for freeze-drying process analytical technology.     

Comparison of heat transfer in liquid and slush nitrogen by numerical simulation of cooling rates for French straws used for sperm cryopreservation

Slush nitrogen (SN₂) is a mixture of solid nitrogen and liquid nitrogen, with an average temperature of −207 °C. To investigate whether plunging a French plastic straw (commonly used for sperm cryopreservation) in SN₂ substantially increases cooling rates with respect to liquid nitrogen (LN₂), a numerical simulation of the heat conduction equation with convective boundary condition was used to predict cooling rates. Calculations performed using heat transfer coefficients in the range of film boiling confirmed the main benefit of plunging a straw in slush over LN₂ did not arise from their temperature difference (−207 vs. −196 °C), but rather from an increase in the external heat transfer coefficient. Numerical simulations using high heat transfer (h) coefficients (assumed to prevail in SN₂) suggested that plunging in SN₂ would increase cooling rates of French straw. This increase of cooling rates was attributed to a less or null film boiling responsible for low heat transfer coefficients in liquid nitrogen when the straw is placed in the solid-liquid mixture or slush. In addition, predicted cooling rates of French straws in SN₂ tended to level-off for high h values, suggesting heat transfer was dictated by heat conduction within the liquid filled plastic straw.     

Local Osmosis and Isotonic Transport

Osmotically driven water flow, u (cm/s), between two solutions of identical osmolarity, c_o (300 mM in mammals), has a theoretical isotonic maximum given by u = j/c_o, where j (moles/cm²/s) is the rate of salt transport. In many experimental studies, transport was found to be indistinguishable from isotonic. The purpose of this work is to investigate the conditions for u to approach isotonic. A necessary condition is that the membrane salt/water permeability ratio, ε, must be small: typical physiological values are ε = 10⁻³ to 10⁻⁵, so ε is generally small but this is not sufficient to guarantee near-isotonic transport. If we consider the simplest model of two series membranes, which secrete a tear or drop of sweat (i.e., there are no externally-imposed boundary conditions on the secretion), diffusion is negligible and the predicted osmolarities are: basal = c_o, intracellular ≈ (1 + ε)c_o, secretion ≈ (1 + 2ε)c_o, and u ≈ (1 − 2ε)j/c_o. Note that this model is also appropriate when the transported solution is experimentally collected. Thus, in the absence of external boundary conditions, transport is experimentally indistinguishable from isotonic. However, if external boundary conditions set salt concentrations to c_o on both sides of the epithelium, then fluid transport depends on distributed osmotic gradients in lateral spaces. If lateral spaces are too short and wide, diffusion dominates convection, reduces osmotic gradients and fluid flow is significantly less than isotonic. Moreover, because apical and basolateral membrane water fluxes are linked by the intracellular osmolarity, water flow is maximum when the total water permeability of basolateral membranes equals that of apical membranes. In the context of the renal proximal tubule, data suggest it is transporting at near optimal conditions. Nevertheless, typical physiological values suggest the newly filtered fluid is reabsorbed at a rate u ≈ 0.86 j/c_o, so a hypertonic solution is being reabsorbed. The osmolarity of the filtrate c_F (M) will therefore diminish with distance from the site of filtration (the glomerulus) until the solution being transported is isotonic with the filtrate, u = j/c_F.With this steady-state condition, the distributed model becomes approximately equivalent to two membranes in series. The osmolarities are now: c_F ≈ (1 − 2ε)j/c_o, intracellular ≈ (1 − ε)c_o, lateral spaces ≈ c_o, and u ≈(1 + 2ε)j/c_o. The change in c_F is predicted to occur with a length constant of about 0.3 cm. Thus, membrane transport tends to adjust transmembrane osmotic gradients toward εc_o, which induces water flow that is isotonic to within order ε. These findings provide a plausible hypothesis on how the proximal tubule or other epithelia appear to transport an isotonic solution.     

哺乳动物毛被传热性能及其影响因素

毛被能够加强或减弱动物向周围环境的热量散失,毛被的形态结构和颜色是传热性能的决定因素,其传热过程往往是传导、对流和辐射3个过程的耦合。以往研究发现环境因子中,风可增加机体向环境中的散热速率,且散失量与风速正相关,且动物通过调节在风场中的姿态来适应不同风向。动物体与环境间的温差是影响散热速率的另一因素,不同环境中的动物通过改变毛被结构来适应温差变化。毛被含水率上升会引起导热和蒸发冷却作用加强,动物通过行为或毛被结构变化来调节毛被含水率。毛色决定毛被吸收和反射热辐射的能力。毛被传热性能直接把动物的生理特点与环境因子关联起来,这对揭示动物的适应、进化都具有重要意义。同时提出,毛被结构和传热性能的研究还有助于仿生学意义的挖掘。因此,今后应重点在毛被结构和物理性能、研究技术与方法以及毛被生物学和仿生学意义等方面开展研究。     

Development of a Mini-Freeze Dryer for Material-Sparing Laboratory Processing with Representative Product Temperature History

The goal of the work described in this publication was to evaluate a new, small, material-sparing freeze dryer, denoted as the “mini-freeze dryer or mini-FD”, capable of reproducing the product temperature history of larger freeze dryers, thereby facilitating scale-up. The mini-FD wall temperatures can be controlled to mimic loading procedures and dryer process characteristics of larger dryers. The mini-FD is equipped with a tunable diode laser absorption spectroscopy (TDLAS) water vapor mass flow monitor and with other advanced process analytical technology (PAT) sensors. Drying experiments were performed to demonstrate scalability to larger freeze dryers, including the determination of vial heat transfer coefficients, K _v . Product temperature histories during K _v runs were evaluated and compared with those obtained with a commercial laboratory-scale freeze dryer (LyoStar II) for sucrose and mannitol product formulations. When the mini-FD wall temperature was set at the LyoStar II band temperature (? 20°C) to mimic lab dryer edge vials, edge vial drying in the mini-FD possessed an average K _v within 5% of those obtained during drying in the LyoStar II. When the wall temperature of the mini-FD was set equal to the central vial product temperature, edge vials behaved as center vials, possessing a K _v value within 5% of those measured in the LyoStar II. During both K _v runs and complete product freeze drying runs, the temperature-time profiles for the average edge vials and central vial in the mini-FD agreed well with the average edge and average central vials of the LyoStar II.     

Behavioral and reproductive responses of female opossums to volatile and nonvolatile components of male suprasternal gland secretion

Female gray short-tailed opossums (Monodelphis domestica) lack an estrous cycle and are induced into estrus by exposure to a pheromone in male scent marks. Behavioral and physiological responses of females to the volatile and nonvolatile components of scent marks were examined in two experiments. Young females (n = 9) were tested prior to and during their first estrus for behavioral responses to scent marks, collected on a 7-ml glass vial rubbed over the suprasternal gland of a mature male. The response to volatile components of the scent mark, recorded when marked and unmarked vials were covered with a perforated shield, was compared to the response to these vials when unshielded. Estrous females nuzzled the shields over marked vials (55.8 ± 8.5 nuzzles/10 min) more than the shielded clean vial (10.9 ± 2.4) (P < 0.05); a similar response was observed in anestrous females. Nuzzling of unshielded, scent-marked vials was higher (P < 0.05) during anestrus than in the same females when in estrus. The role of nonvolatile pheromones in reproductive activation was tested in adult females (n = 11) exposed for up to 14 days to a shielded, marked vial or to an unshielded, marked vial in a crossover design. All females exposed to unshielded vials expressed estrus, and 10 copulated. Only 2 females expressed estrus (significantly fewer, P < 0.05), when exposed to shielded marked vials, and neither copulated. These results demonstrate that females detect and respond behaviorally to both volatile and nonvolatile components of male suprasternal gland secretion, but the estrus-inducing pheromone in these secretions is nonvolatile.     

Evaluation of manometric temperature measurement, a process analytical technology tool for freeze-drying: Part I, product temperature measurement

This study examines the factors that may cause systematic errors in the manometric temperature measurement (MTM) procedure used to evaluate product temperature during primary drying. MTM was conducted during primary drying using different vial loads, and the MTM product temperatures were compared with temperatures directly measured by thermocouples. To clarify the impact of freeze-drying load on MTM product temperatures, simulation of the MTM vapor pressure rise was performed, and the results were compared with the experimental results. The effect of product temperature heterogeneity in MTM product temperature determination was investigated by comparing the MTM product temperatures with directly measured thermocouple product temperatures in systems differing in temperature heterogeneity. Both the simulated and experimental results showed that at least 50 vials (5 mL) were needed to give sufficiently rapid pressure rise during the MTM data collection period (25 seconds) in the freeze dryer, to allow accurate determination of the product temperature. The product temperature is location dependent, with higher temperature for vials on the edge of the array and lower temperature for the vials in the center of the array. The product temperature heterogeneity is also dependent upon the freeze-drying conditions. In product temperature heterogeneous systems, MTM measures a temperature close to the coldest product temperature, even, if only a small fraction of the samples have the coldest product temperature. The MTM method is valid even at very low product temperature (−45°C). Published: February 10, 2006     

Cross-contamination during lyophilization

Cross-contamination during lyophilization was studied in two freeze-dryers. Suspensions of test organisms (Pseudomonas aeruginosa and Proteus mirabilis) sensitive to colistin and neomycin, respectively, were dispensed into vials, freeze-dried, and plated on an agar medium containing colistin or neomycin.In a small, chambered freeze-dryer, 0.8–3.3% cross-contamination occurred when each side of the tray had a different organism, and 8.3–13.3% when a cluster of vials in the middle was surrounded by vials with the other organism. Preliminary results in a larger freeze-dryer showed no shelf-to-shelf cross-contamination.Since the condenser port of the smaller freeze-dryer was underneath the shelf holding the vials during sublimation, and the larger freeze-dryer had a port in the rear wall of the chamber, cross-contamination may be dictated (at least partially) by this condition.     

An economical system for liquid scintillation counting

Small glass shell vials (12 × 35 mm minivials), containing 2.0 ml of a dioxane-based scintillation solution plus a ¹⁴C-labeled sample, were placed in a conventional glass, 20-ml count vial and assayed in a scintillation spectrometer. Statistical comparison of counts recorded from ¹⁴C samples prepared both in the minivial system and conventional 20-ml count vials indicated that the two systems were equivalent with sample volumes of 10 and 100 μliters (1600-cpm solution) and 10 μliters (60-cpm solution). Conventional 20-ml glass or plastic count vials were both acceptable as containers for the minivials.There were no significant differences in the counts from samples in minivials placed on-center and off-center in the container vial. Cost per sample was reduced from 24.8 cents (conventional glass vials) to 4.7 cents (minivial system).     

Macrofaunal abundance at 79°N off East Greenland: opposing data from epibenthic-sledge and box-corer samples

During the expedition ARK XI-2 with RV Polarstern in September/October 1995, a transect of samples was taken off East Greenland from the shelf down to about 1800 m depth by means of an epibenthic sledge and a giant box corer, in order to obtain information on epibenthic and endobenthic macrofauna. Within the epibenthic sledge catches, the Cumacea were the most numerous taxon with 32,123 (always calculated for 1000 m) specimens, followed by the Isopoda with 26,914 specimens, and Amphipoda with 20,900; Mysidacea and Tanaidacea were less important. Amphipoda were the most numerous taxon on the shelf, while with increasing depth Isopoda and especially Cumacea became increasingly important. Macrofauna analysed from the box-corer samples decreased in abundance with depth. The highest macrofaunal abundance was found on the shelf, with 8128 ind. m⁻² (200 m); below this depth, density decreased from ≈2000 ind. m⁻² (at 800 m) to ≈800 ind. m⁻² (2000 m). Peracarid crustaceans and polychaetes were, next to bivalves, the most frequent. Polychaetes clearly dominated the shelf communities; their relative percentage decreased with increasing water depth while the relative percentage of Peracarida increased simultaneously. Interestingly, the highest number of peracarids in the epibenthic-sledge samples was reported from deep sea, while box-corer samples showed decreasing macrofaunal abundance with depth. The relative percentage of peracarids in the box-corer samples increased with depth, while in absolute numbers, peracarids from these samples were also highest on the shelf at about 200 m depth. Accepted: 14 February 1999     

Transient promoter activity in primary rat mammary epithelial cells evaluated using particle bombardment gene transfer

Summary The relative strengths of several commonly used viral promoters in primary cultures of rat mammary epithelial cells were studied using a particle bombardment gene transfer method. NIH 3T3 cells were also examined as a representative cell line. Initially, the conditions necessary for efficient gene transfer using particle bombardment were determined. Discharge voltage for particle bombardment was evaluated to maximize the levels of gene expression and cell viability. After transfection, transgene expression decreased over a 5-day period in both mammary cells and NIH 3T3 cells. Particle bombardment gene transfer was at least fivefold more efficient than lipofection, calcium phosphate co-precipitation, or electroporation. The activity of five viral enhancer/promoters was compared using a luciferase gene assay system. The relative promoter strengths in mammary cells were determined to be: RSV ≈ CMV ≈ SV40 > MLV > MMTV. Tissue-specific activity of the MMTV-LTR was demonstrated, although this promoter conferred the lowest expression level among the promoters tested.     

Behavioral and reproductive responses of female opossums to volatile and nonvolatile components of male suprasternal gland secretion

Female gray short-tailed opossums (Monodelphis domestica) lack an estrous cycle and are induced into estrus by exposure to a pheromone in male scent marks. Behavioral and physiological responses of females to the volatile and nonvolatile components of scent marks were examined in two experiments. Young females (n = 9) were tested prior to and during their first estrus for behavioral responses to scent marks, collected on a 7-ml glass vial rubbed over the suprasternal gland of a mature male. The response to volatile components of the scent mark, recorded when marked and unmarked vials were covered with a perforated shield, was compared to the response to these vials when unshielded. Estrous females nuzzled the shields over marked vials (55.8 ± 8.5 nuzzles/10 min) more than the shielded clean vial (10.9 ± 2.4) (P < 0.05); a similar response was observed in anestrous females. Nuzzling of unshielded, scent-marked vials was higher (P < 0.05) during anestrus than in the same females when in estrus. The role of nonvolatile pheromones in reproductive activation was tested in adult females (n = 11) exposed for up to 14 days to a shielded, marked vial or to an unshielded, marked vial in a crossover design. All females exposed to unshielded vials expressed estrus, and 10 copulated. Only 2 females expressed estrus (significantly fewer, P < 0.05), when exposed to shielded marked vials, and neither copulated. These results demonstrate that females detect and respond behaviorally to both volatile and nonvolatile components of male suprasternal gland secretion, but the estrus-inducing pheromone in these secretions is nonvolatile.     

A thermal model of the human body exposed to an electromagnetic field

The human body was modeled by numerical procedures to determine the thermal response under varied electromagnetic (EM) exposures. The basic approach taken was to modify the heat transfer equations for man in air to account for thermal loading due to the energy absorbed from the EM field. The human body was represented in an EM model by a large number of small cubical cells of tissue, and the energy density was determined for each cell. This information was then analyzed by a thermal response model consisting of a series of two-dimensional transient conduction equations with internal heat generation due to metabolism, internal convective heat transfer due to blood flow, external interaction by convection and radiation, and cooling of the skin by sweating and evaporation. This model simulated the human body by a series of cylindrical segments. The local temperature at 61 discrete locations as well as the thermoregulatory responses of vasodilatation and sweating were computed for a number of EM field intensities and two frequencies, one near whole-body resonance.     

Effects of body-size and season on digestive organ size and the energy balance of cockles fed with a constant diet of phytoplankton

Seasonal variation in size-dependence of seawater clearance rate, absorption efficiency, oxygen consumption, gill area, length of the crystalline style and dry weight of digestive gland was analyzed in cockles Cerastoderma edule from the Mundaka Estuary, Spain. Experimental determinations were performed monthly (from July 1998 to November 1999) in cockles being fed with Tetraselmis suecica (organic content: 87.84 ± 1.95%) at a concentration of 3 mm³/l for 3 days. Analysis of covariance reveals no seasonal differences in both size-dependence of seawater clearance rate and oxygen consumption, which were found to scale to dry body weight with mass-exponents of 0.56 and 0.62, respectively. No significant correlation was found between absorption efficiency and body weight. Mass-exponents for gill area, dry weight of the digestive gland and length of the crystalline style remained constant among seasons showing values of 0.62, 0.34 and 0.82, respectively. Seasonal trends for every physiological determination were calculated for a standard size (200 mg) cockle: standardized clearance rates and oxygen consumptions followed a similar trend with minimum values in winter (≈0.5 l/h and ≈100 μl O₂/h, respectively) and maximum values during spring–summer (≈1.7 l/h and ≈250 μl O₂/h, respectively), whereas absorption efficiency and food throughput time showed both the opposite pattern with highest values corresponding to winter months (≈50–60% and ≈5–6 h, respectively), and lowest (≈30% and ≈3–4 h, respectively) to summer–autumn. Scope for growth exhibited minimum values in winter followed by a rapid increase along the winter–spring transition, maximum values being attained in spring (May) and summer (July). Exponential decline of seasonal values of absorption efficiency associated to rising ingestion rates of organic matter presented an asymptotic minimum at 0.35. Absorption efficiency was positively related to food throughput time, whereas the latter fell to a minimum of 3.548 h with increasing food intake. So, maintenance of throughput time—and consequently absorption efficiency—along with enhanced filtering activity provided cockles with higher absorption rates improving scopes for growth registers during spring and summer. These dynamics might be explained as the consequence of the seasonal digestive adjustments in cockles, which, in fact, were found to increase the size of the digestive organs during that period.     

Entropy budgets of deciduous plant leaves and a theorem of oscillating entropy production

From an energy budget of a deciduous plant leaf in moderate conditions, entropy fluxes into or out of the leaf due to solar radiation, infrared radiation, evaporation of water and heat conduction are calculated. Net entropy flow into the leaf is negative. On the assumption that the entropy in the leaf is in a steady state, the entropy production in the typical deciduous leaf in moderate conditions [the solar energy absorbed by both sides of the leaf isE _solar=0.0602 (J cm⁻² s⁻¹)] becomesS _prod=1.8×10⁻⁴ (J cm⁻² s⁻¹ K⁻¹). The positiveness of the entropy production shows that the Second Law of Thermodynamics certainly holds in the plant leaf. Entropy productions in other conditions are also calculated. The entropy production in the leafS _prod becomes a linear function of the solar energy absorbed by the leafE _solar:S _prod≈-(29.5E _solar)×10⁻⁴. A theorem is presented: the entropy production in plant leaves oscillates during the period of one day, paralleling the daily solar energy absorbed by leaves.