Saturation Constant Research Articles

Indirect bottom‐up control of consumer‐resource dynamics: Resource‐driven algal quality alters grazer numerical response

The nature of the growth‐rate response of grazing zooplankton to algal concentration depends critically upon food quality, which is often a plastic function of environmental conditions. Phytoplankton quality depends upon nutrient and light availability, which together affect cell C : P and C : N, so that the links from external environment to phytoplankton and then to zooplankton can vary in ways that influence consumer‐resource interaction strengths, and ultimately, food web structure. We measured the effects of resource‐limitation on the population growth rate of rotifers, Brachionus calyciflorus, feeding on the alga Chlamydomonas reinhardtii, and grown in nitrogen‐limiting, phosphorus‐limiting, or nutrient‐sufficient conditions. Consumption of nutrient‐sufficient algae produced a Type II numerical response with a low half saturation constant (Ks) and high asymptotic growth rate (λmax) consistent with high food quality; P‐limited algae also yielded a Type II response but with high Ks and low λmax, meaning poor food quality. In contrast, N‐limited Chlamydomonas resulted in a novel Type IV numerical response in which rotifer growth rate rose to an intermediate λmax and then declined at the highest algal densities, suggesting the production of toxic compounds whose negative effect on growth increases as a function of cell density. Experiments with mixed diets confirmed that N‐limited food was indeed toxic, whereas P‐limited food was simply nutritionally deficient compared with light‐limited algae. Additional experiments showed that even the filtrate of N‐limited algae can be detrimental to rotifer growth.

SATURATION EFFECTS FOR CTL MEDIATED CONTROL OF HIV-1 INFECTION: A MATHEMATICAL STUDY

The relations between the Human Immunodeficiency Virus-1 (HIV-1) and the human immune system are astonishingly multifaceted, where the critical role for cytotoxic T lymphocytes (CTLs) in the suppression of viral replication in HIV-1 infected individuals cannot be ignored. In this research paper, we have proposed a mathematical model incorporating half saturation constant through the CTL mediated killing process and also in that sense, it has been infiltrated in the generation process of CTL through infected cells. To make the model more realistic, a time lag is introduced in the generation term of CTL population. Also an optimal control theory paradigm is used in our mathematical model to suppress the viral production. From our entire analysis, we have found threshold condition of half saturation constant and treatment schedule so that we can handle the situation of Acquired Immunodeficiency Syndrome (AIDS) patients in a better way. Our analysis reveals that, if the half saturation constant is around 47 mm-3in the saturation process and the drug therapy is to be used around 76 days, then we can get adequate results for better treatment of a HIV-1 patient. Based on numerical results, we observed that in a highly unstable situation, administration of chemotherapy at a high dose can stabilize the system.

Determination of kinetic parameters in a pilot scale BNR system treating municipal wastewater

Kinetics for the biological processes of nitrification, denitrification and carbon oxidation were studied in the aerobic and anoxic phases of a pilot scale Biological Nutrient Removal (BNR) plant treating municipal wastewater. The configuration of the treating system is based on the combination of the UCT (University of Cape Town) design and the step feeding process in a cascade denitrification. In order to study the process kinetics and to obtain reliable values for the investigated kinetic parameters batch experiments were performed. For this purpose, continuous feeding of the treating system was interrupted for a given period of time and the pilot plant was turned into a batch mode of operation. Thereafter, addition of NO3 --N and NH4 +-N into the anoxic and aerobic compartments of the treating plant, respectively, followed, whereas adequate initial concentration of a carbon source (municipal wastewater or synthetic substrate) was ensured in the mixed liquor. Experimental data indicated that the examined biological processes followed saturation kinetics. The maximum specific denitrification rate, qDN,max , was found to obtain values, depending on the type of the carbon source, between 0,045 and 0,390 gNO3 --N/(gXHET·d), whereas the extremely low value of the half saturation constant for the denitrification process (Km,NO3-N << 1mgN/l) indicated its description by zero order kinetics. The maximum specific nitrification rate, qN,max, was determined to vary in a narrow frame, between 1,28 and 1,60 gNH4 +- N/(gXAO·d). The half saturation constant for the nitrification process, Km,NH4-N, was estimated graphically at 3,1 – 6,1 gNH4 +-N l-1, corresponding to 62 – 122 μgNH3-N l-1. These values are considered to be in good agreement with the literature. The determination of kinetic parameters can be considered as a useful tool for the process design, operation and improvement of wastewater treatment plants. Furthermore, the study of the biological process kinetics contributes to the better understanding and outline of the complicated biological processes that contemporarily take place within the various phases of BNR wastewater treatment plants.

Downscaling the marine modelling effort: Development, application and assessment of a 3D ecosystem model implemented in a small coastal area

The present study deals with the development, application and evaluation of a modelling tool, implemented along with a field sampling program, in a limited coastal area in the Northeast Aegean. The aim was to study, understand and quantify physical circulation and water column ecological processes in a high resolution simulation of a past annual cycle. The marine ecosystem model consists of a three dimensional hydrodynamic component suitable for coastal areas (Princeton Ocean Model) coupled to a simple ecological model of five variables, namely, phytoplankton, nitrate, ammonia, phosphate and dissolved organic carbon concentrations. The ecological parameters (e.g. half saturation constants and maximum uptake rates for nutrients) were calibrated using a specially developed automated procedure. Model errors were evaluated using qualitative, graphic techniques and were quantified with a number of goodness-of-fit measures. Regarding physical variables, the goodness-of-fit of model to field data varied from fairly to quite good. Indicatively, the cost function, expressed as mean value per sampling station, ranged from 0.15 to 0.23 for temperature and 0.81 to 3.70 for current speed. The annual cycle of phytoplankton biomass was simulated with sufficient accuracy (e.g. mean cost function ranging from 0.49 to 2.67), partly attributed to the adequate reproduction of the dynamics of growth limiting nutrients, nitrate, ammonia and the main limiting nutrient, phosphate, whose mean cost function ranged from 0.97 to 1.88. Model results and field data provided insight to physical processes such as the development of a wind-driven, coastal jet type of surface alongshore flow with a subsurface countercurrent flowing towards opposite direction and the formation of rotational flows in the embayments of the coastline when the offshore coastal current speed approaches values of about 0.1 m/s. The percentage of field measurements where the N:P ratio was found over 16:1 varied between stations from 57 to 65%, demonstrating the importance of phosphate as a limiting nutrient for phytoplankton growth. The model also successfully reproduced phytoplankton gradients, e.g. those developed almost all year round between the most eutrophic areas (city harbour with mean chlorophyll-a concentration of 1.08 μg/L and decreasing biomass by over 40 mg C/m3 from surface to the bottom layer) and the oligotrophic open waters (mean chlorophyll-a concentration of 0.14 μg/L).

Kinetics and mechanisms of hydrogen sulfide adsorption by biochars

Three different biochars as cost-effective substitutes for activated carbon (AC) were tested for their hydrogen sulfide (H2S) adsorption ability. The biochars were produced from camphor (SC), bamboo (SB), and rice hull (SR) at 400°C by oxygen-limited pyrolysis. The surface area (SA), pH, and Fourier transform infrared spectras of the biochars and AC were compared. The maximum removal rates and the saturation constants were obtained using the Michaelis–Menten-type equation. The three biochars were found to be alkaline, and the SAs of the biochars were much smaller than that of the AC. The H2S breakthrough capacity was related to the local pH within the pore system of the biochar. The order observed in terms of both biochar and AC adsorption capacity was SR>SB>SC>AC. SR efficiently removed H2S within the inlet concentration range of 10–50μL/L. Biochars derived from agricultural/forestry wastes are a promising H2S adsorbent with distinctive properties.

Biochemical Kinetics of Cross flow Membrane Bioreactor Processes in the Treatment of Refinery Wastewater

A lab-scale cross flow membrane bioreactor (CF-MBR) was operated to determine the biokinetic coefficients under MLSS concentrations of 5000 and 3000 mg/L. The investigation showed that the yield (Y), the endogenous decay coefficient (kd), the maximum specific growth rate (μm) and the saturation constant (KS) were 0.276 mg/mg, 0.07 /day, 0.653 /day, and 396.62 mg COD/L respectively for MLSS 5000 mg/L, and 0.222 mg/mg, 0.09 /day, 1.2 /day, and 659.45 mg COD/L for MLSS 3000 mg/L. The values of kinetic coefficients were within the normal range of the activated sludge process found in the literature, except the values of Y. However, value of Y increased with the increase of MLSS. Kinetic parameters determined from CF-MBR process were used to simulate the effluent COD. The simulation study showed good agreement between model prediction and experimental data. Sensitivity analysis was carried out to determine influence of biokinetic parameters on the effluent substrate concentration. From the analysis, it was evident that kd and KS were directly proportional to the effluent substrate concentration, while μm was inversely proportional.

Steady State Performance of a Bioreactor for Production of near Zero Sulfur Diesel (NZSD) and Bio-surfactant

Abstrac t—Kinetics of biodesulfurization of hydrotreated diesel using Rhodococcus sp has been studied with special reference to removal of organo-sulfur compounds in diesel and production of 2-hydroxy biphenyl. The identification of 2-HBP in treated diesel has been made using HPLC,FTIR and GCMS .The sulfur concentration of feed diesel was in the range of 200-540 mg/L. Aqueous phase to diesel ratios were varied in the range of 9:1 to1:9. The optimum ratio was found to be 1:4 and the maximum conversion of sulfur was determined to be 95%. The values of Monod kinetic parameters namely, µ m ax, maximum specific gro wth rate and Ks, saturation constant of the microbial growth and Yield coefficient of surfactant have been measured to be 0.096h -1 , 71mg/L , and 17 µmol/g dry cell weights respectively by conducting batch type experiments. A continuous chemostat was studied using different hydrodynamic and physico-chemical parameters like dilution rate, initial concentration of organo-sulfur compounds in diesel, stirring rate and aeration rate. Surfactant part was characterized by determination of surface tension (Ring method), E24, TLC, HPLC and GC-MS. The interfacial tension of the supernatant fermented by Rhodococcus sp decreased from 28 dynes/cm to 9 dynes/cm.The surface tension of aqueous nutrient phase and diesel was observed to decrease from 71 dynes/cm to 30 dynes/cm and to 20 dynes/cm from 30dynes/cm respectively. Values of emulsification index (E24) were determined to vary from 18 to 58 over the growth period of 2 to 48 hours in the chemostat. The critical miceller concentration was found to be 200mg/L. The simulated data have been compared with the experimental ones to validate the model.

Cross-Diffusion-Driven Instability in a Reaction-Diffusion Harrison Predator-Prey Model

We present a theoretical analysis of processes of pattern formation that involves organisms distribution and their interaction of spatially distributed population with cross-diffusion in a Harrison-type predator-prey model. We analyze the global behaviour of the model by establishing a Lyapunov function. We carry out the analytical study in detail and find out the certain conditions for Turing’s instability induced by cross-diffusion. And the numerical results reveal that, on increasing the value of the half capturing saturation constant, the sequences “spots → spot-stripe mixtures → stripes → hole-stripe mixtures → holes” are observed. The results show that the model dynamics exhibits complex pattern replication controlled by the cross-diffusion.

Kinetic Studies of Alkaline Protease from Bacillus licheniformis NCIM-2042

An extensive investigation was carried out to describe the kinetics of cell growth, substrate consumption, and product formation in the batch fermentation using starch as substrate. Evaluation of intrinsic kinetic parameters was carried out using a best-fit unstructured model. A nonlinear regression technique was applied for computational purpose. The Andrew's model showed a comparatively better R2 value among all tested models. The values of specific growth rate (micronmax), saturation constant (KS), inhibition constant (KI), and YX/S were found to be 0.109 h-1, 11.1 g/l, 0.012 g/l, and 1.003, respectively. The Leudeking- Piret model was used to study the product formation kinetics and the process was found to be growth-associated. The growth-associated constant (alpha) for protease production was sensitive to substrate concentration. Its value was fairly constant up to a substrate concentration of 30.8 g/l, and then decreased.

Isolation and characterization of bacterial strains that have high ability to degrade 1,4-dioxane as a sole carbon and energy source

Four novel metabolic 1,4-dioxane degrading bacteria possessing high ability to degrade 1,4-dioxane (designated strains D1, D6, D11 and D17) were isolated from soil in the drainage area of a chemical factory. Strains D6, D11 and D17 were allocated to Gram-positive actinomycetes, similar to previously reported metabolic 1,4-dioxane degrading bacteria, whereas strain D1 was allocated to Gram-negative Afipia sp. The isolated strains could utilize a variety of carbon sources, including cyclic ethers, especially those with carbons at position 2 that were modified with methyl- or carbonyl-groups. The cell yields on 1,4-dioxane were relatively low (0.179-0.223mg-protein (mg-1,4-dioxane)(-1)), which was likely due to requiring energy for C-O bond fission. The isolated strains showed 2.6-13 times higher specific 1,4-dioxane degradation rates (0.052-0.263mg-1,4-dioxane (mg-protein)(-1) h(-1)) and 2.3-7.8 fold lower half saturation constants (20.6-69.8mgL(-1)) than the most effective 1,4-dioxane degrading bacterium reported to date, Pseudonocardia dioxanivorans CB1190, suggesting high activity and affinity toward 1,4-dioxane degradation. Strains D1 and D6 possessed inducible 1,4-dioxane degrading enzymes, whereas strains D11 and D17 possessed constitutive ones. 1,4-Dioxane degradation (100mgL(-1)) by Afipia sp. D1 was not affected by the co-existence of up to 3,000mgL(-1) of ethylene glycol. The effects of initial pH, incubation temperature and NaCl concentration on 1,4-dioxane degradation by the four strains revealed that they could degrade 1,4-dioxane under a relatively wide range of conditions, suggesting that they have a certain adaptability and applicability for industrial wastewater treatment.

Modelling the Effects of Competing Toxin Producing Phytoplankton on a Zooplankton Population: Role of Holling Type-II Functional Response with Time-Lag

In this paper, a system consisting of two competing harmfu l phytoplankton and a zooplankton with Holling type-II functional response and discrete time lag is considered. A stable co-existence of all the species has been observed for the system without delay and the Hopf-b ifurcation phenomenon is observed for the interior equilibriu m po int. The Hopf-bifurcating solution is obtained for the critical values of parameters like predation rates and half saturation constants. Further, using the normal form theory, we have determined the direction and the stability of the Hopf-bifurcation solution. The introduction of time delay in the system also shows the Hopf-bifurcat ion as the delay parameter passes through a critical value. Finally, the numerical simu lation is carried out to support the theoretical results.

Study on the aerobic biodegradability and degradation kinetics of 3-NP; 2,4-DNP and 2,6-DNP

Four biodegradability tests (BOD5/COD ratio, production of carbon dioxide, relative oxygen uptake rate and relative enzymatic activity) were used to determine the aerobic biodegradability of 3-nitrophenol (3-NP), 2,4-dinitrophenol (2,4-DNP) and 2,6-dinitrophenol (2,6-DNP). Furthermore, biodegradation kinetics of the compounds was investigated in sequencing batch reactors both in the presence of glucose (co-substrate) and with nitrophenol as the sole carbon source. Among the three tested compounds, 3-NP showed the best biodegradability while 2,6-DNP was the most difficult to be biodegraded. The Haldane equation was applied to the kinetic test data of the nitrophenols. The kinetic constants are as follows: the maximum specific degradation rate (Kmax), the saturation constants (KS) and the inhibition constants (KI) were in the range of 0.005–2.98mg(mgSSd)−1, 1.5–51.9mgL−1 and 1.8–95.8mgL−1, respectively. The presence of glucose enhanced the degradation of the nitrophenols at low glucose concentrations. The degradation of 3-NP was found to be accelerated with the increasing of glucose concentrations from 0 to 660mgL−1. At high (1320–2000mgL−1) glucose concentrations, the degradation rate of 3-NP was reduced and the Kmax of 3-NP was even lower than the value obtained in the absence of glucose, suggesting that high concentrations of co-substrate could inhibit 3-NP biodegradation. At 2,4-DNP concentration of 30mgL−1, the Kmax of 2,4-DNP with glucose as co-substrate was about 30 times the value with 2,4-DNP as sole substrate. 2,6-DNP preformed high toxicity in the case of sole carbon source degradation and the kinetic data was hardly obtained.

The kinetics of denitrification in permeable sediments

Permeable sediments comprise the majority of shelf sediments, yet the rates of denitrification remain highly uncertain in these environments. Computational models are increasingly being used to understand the dynamics of denitrification in permeable sediments, which are complex environments to study experimentally. The realistic implementation of such models requires reliable experimentally derived data on the kinetics of denitrification. Here we undertook measurements of denitrification kinetics as a function of nitrate concentration in carefully controlled flow through reactor experiments on sediments taken from six shallow coastal sites in Port Phillip Bay, Victoria, Australia. The results showed that denitrification commenced rapidly (within 30 min) after the onset of anoxia and the kinetics could be well described by Michaelis–Menten kinetics with half saturation constants (apparent Km) ranging between 1.5 and 19.8 μM, and maximum denitrification rate (Vmax) were in the range of 0.9–7.5 nmol mL−1 h−1. The production of N2 through anaerobic ammonium oxidation (anammox) was generally found to be less than 10 % of denitrification. Vmax were in the same range as previously reported in cohesive sediments despite organic carbon contents one order of magnitude lower for the sediments studied here. The ratio of sediment O2 consumption to Vmax was in the range of 0.02–0.09, and was on average much lower than the theoretical ratio of 0.8. As a consequence, models implemented with the theoretical ratio of 0.8 are likely to overestimate denitrification by a factor of ~3. The most likely explanation for this is that the microbial community is not able to instantaneously shift or optimally use a particular electron acceptor in the highly dynamic redox environment experienced in permeable sediments. In contrast to previous studies, we did not observe any significant rates of oxic denitrification.

Removal of imidazolium ionic liquids by microbial associations: Study of the biodegradability and kinetics

The aim of this study was to estimate the biodegradability of the selected imidazolium ionic liquids and to determine the kinetic parameters for the biological treatment of wastewater containing these ionic liquids. Biodegradability was evaluated with the help of the Organisation for Economic Co-operation and Development (OECD) tests, while oxygen uptake rate (OUR) tests were made in order to calculate the values of Monod kinetic parameters. The results obtained from both types of the tests showed that ionic liquids of chemical structure of 1-alkyl-3-methyl imidazolium bromide were poorly biodegradable and co-biodegradable compounds, although their biodegradability increased with the elongation of the alkyl chain length. At the same time the presence of the imidazolium-based ionic liquids in wastewater at concentration of 50 mg l(-1) did not inhibit biomass growth as well as did not decrease the affinity of substrate to biomass. The values of both Monod kinetic parameters, i.e., maximum specific growth rate (μ(max)) and half saturation constant (K(S)), increased with the increase in chain length of the alkyl substituent.

Effect of carbon source during enrichment on BTEX degradation by anaerobic mixed bacterial cultures

A comprehensive study on the effects of different carbon sources during the bacterial enrichment on the removal performances of benzene, toluene, ethylbenzene, and xylenes (BTEX) compounds when present as a mixture was conducted. Batch BTEX removal kinetic experiments were performed using cultures enriched with individual BTEX compounds or BTEX as a mixture or benzoate alone or benzoate-BTEX mixture. An integrated Monod-type non-linear model was developed and a ratio between maximum growth rate (μ max) and half saturation constant (Ks) was used to fit the non-linear model. A higher μ max/Ks indicates a higher affinity to degrade BTEX compounds. Complete removal of BTEX mixture was observed by all the enriched cultures; however, the removal rates for individual compounds varied. Degradation rate and the type of removal kinetics were found to be dependent on the type of carbon source during the enrichment. Cultures enriched on toluene and those enriched on BTEX mixture were found to have the greatest μ max/Ks and cultures enriched on benzoate had the least μ max/Ks. Removal performances of the cultures enriched on all different carbon sources, including the ones enriched on benzoate or benzoate-BTEX mixture were also improved during a second exposure to BTEX. A molecular analysis showed that after each exposure to the BTEX mixture, the cultures enriched on benzoate and those enriched on benzoate-BTEX mixture had increased similarities to the culture enriched on BTEX mixture.

Inhibition Kinetics of Ammonia Oxidation Influenced by Silver Nanoparticles

Silver nanoparticles (AgNPs) have significantly increased in production and use for anti-microbial propose. This agent, after used, is released into sewerage system resulting in possibility to inactivate non-targeted microorganisms in wastewater treatment plants. In this study, the inhibitory effect of AgNPs on ammonia oxidation was investigated using respirometric assay. The initial concentrations of AgNPs and ammonia ranged 0.25–10.00 and 14–280 mg/L, respectively. Half saturation constant (Ks) for ammonia oxidation was found to be 15.9 mg N/L. Under the presence of AgNPs, the maximum oxygen uptake rate and Ks declined. The effect of AgNPs was proved to follow an uncompetitive-like inhibition kinetic type with the inhibition coefficients (Ki) of 5.5 mg/L. Increasing AgNPs from 0.25 to 10.00 mg/L inhibited 4 to 50 % of ammonia-oxidizing activities at the initial ammonia concentrations from 14 to 280 mg/L. Based on transmission electron microscopic observation, AgNPs could damage the microbial cells. All findings indicated that AgNPs substantially reduced ammonia-oxidizing microorganisms and their activities. Thus, special attention should be made to manage discharge of AgNPs into the environment.

Effect of pH, substrate and free nitrous acid concentrations on ammonium oxidation rate

Respirometric techniques have been used to determine the effect of pH, free nitrous acid (FNA) and substrate concentration on the activity of the ammonium oxidizing bacteria (AOB) present in an activated sludge reactor. With this aim, bacterial activity has been measured at different pH values (ranging from 6.2 to 9.7), total ammonium nitrogen concentrations (ranging from 0.1 to 10mgTANL−1) and total nitrite concentrations (ranging from 3 to 43mgNO2–NL−1). According to the results obtained, the most appropriate kinetic expression for the growth of AOB in activated sludge reactors has been established. Substrate half saturation constant and FNA and pH inhibition constants have been obtained by adjusting model predictions to experimental results. Different kinetic parameter values and different Monod terms should be used to model the growth of AOB in activated sludge processes and SHARON reactors due to the different AOB species that predominate in both systems.

Exergy as an indicator: Observations of an aquatic ecosystem model

Exergy is considered as a goal function or ecological orientor. Normally at the edge of oscillation exergy reaches to its maximum value when the ecosystem had no adaptation on it. To study the variation of exergy in different states of ecosystem, a simple three species (phytoplankton, zooplankton and fish) food chain model has been considered. From the model it is shown that the system moves from steady state to chaotic state by decreasing zooplankton body volume in turn increasing its grazing rate. By the property of self-adaptability the system tries to overcome this situation. Two such possible processes are described here: (i) by the toxic effect of phytoplankton and (ii) by reducing half saturation constant of fishes. In both this cases exergy value reduces and the system reaches to stable state. Through the analysis of exergy variation in all these situations this paper shows that the system chose the process in which the reduction of exergy will be the minimum.

Methylene Blue Degradation by Sphingomonas paucimobilis under Aerobic Conditions

The presence of synthetic dyes in industrial wastewaters may create serious environmental problems due to their mutagenicity and toxicity to aquatic life and humans. In this study, the decolourization and degradation of methylene blue (MB) by a Sphingomonas paucimobilis strain isolated from industrial wastewater was investigated under aerobic conditions. Decolourization extent of MB in medium was over 85 % when the bacterium was grown on a high concentration of the dye (1,000 mg/L) after a retention time of 5 days, while reduction in COD was 92.99 % suggesting mineralization of dyes as a result of microbial activities. The bacterium retained decolourizing activity over a wide range of pH (2–10), with peak activity obtained at pH 9. Analysis of samples extracted from decolourized culture flasks at pH 9 using UV–visible and Fourier transform infrared (FTIR) spectroscopy confirmed that the mechanism of colour removal was due to biodegradation rather than adsorption of dye on cells. Scanning and transmission electron microscopy revealed the secretion of exopolysaccharides (EPS) by S. paucimobilis cells on exposure to MB—a probable physiological defence mechanism to ensure controlled diffusion of dye molecules into cellular structures. Biokinetic coefficients, namely, growth yield, Y; specific biomass decay, Kd; maximum specific substrate rate, k; saturation constant for substrate, Ks; and maximum specific biomass growth rate, μmax, were determined by the Monod type kinetic equation. Results indicate that S. paucimobilis holds a promise as a good candidate for the biological treatment of industrial effluent containing high concentrations of synthetic dyes.

Horizontal flow and capillarity-driven redistribution in porous media

A recent macroscopic mixture theory for two-phase immiscible displacement in porous media has introduced percolating and nonpercolating phases. Quasi-analytic solutions are computed and compared to the traditional theory. The solutions illustrate physical insights and effects due to spatiotemporal changes of nonpercolating phases, and they highlight the differences from traditional theory. Two initial and boundary value problems are solved in one spatial dimension. In the first problem a fluid is displaced by another fluid in a horizontal homogeneous porous medium. The displacing fluid is injected with a flow rate that keeps the saturation constant at the injection point. In the second problem a horizontal homogeneous porous medium is considered which is divided into two subdomains with different but constant initial saturations. Capillary forces lead to a redistribution of the fluids. Errors in the literature are reported and corrected.