Pressure probes

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INTRODUCTION:

Pressure probes employs the pressure measured at the flow velocity which have been employed to define fluid flows and measuring mean or average velocity is to use a pitot static tube. Pressure probes can be used measure the stagnation pressure, the static pressure, and the flow angle within a fluid stream. When designing a pneumatic probe that is to be used in flow measurements, the effects of blockage, frequency response, pressure hole size and geometry, the local Mach and Reynolds numbers and the relative scale of the phenomena under investigation must be addressed. In general, better accuracy is obtainable if smaller probes and transducers are used though this usually means that the mechanical integrity may be compromised, that the response times are longer and that there are greater problems with contamination in dirty environments. The effects of blockage become greatest when the flow is compressible. For example, inserting a probe into a gas stream where the undisturbed Mach number is 0.9 will cause the flow to choke if the flow area is reduced by only 1 percent. In supersonic flow, a probe stem can create a detached bow shock that lies far upstream of the probe tip.

Chu and Young originally proposed and demonstrated the use of straight tube with its end cut at an oblique angle to the tube axis applied to a study of the flow near a wing body junction. Chu's procedure requires measurement of the pressure at a series of four angles of the tube axis (roll angles) for every 90° intervals and also employed the use of a square-cut sliding cover was extended over the rotatable to provide fifth pressure measurement.

Explicit calibration of the straight tube required fitting of the response to several polynomials and involves dozens of calibration constants. The five pressure measurements are used to employ five-hole probe to determine velocity magnitude and direction. Bryer and Pankhurst provide a good summary of accuracy penalties incurred for misalignment for different pitot static tube design. Dynamic and Static pressures are required for complex flows and it's achieved by multi-hole pressure probe which typically requires calibration. Each measured pressure for rotations of the tube was obtained from one of the four circumferential holes of a 5-hole probe and pressure measured with the cover extended was used from the central hole of a 5-hole probe.

Straight tube with oblique cut end have many potential advantages when compared to multi-hole probes and most obvious may be its size for a given tube diameter, a five hole probe would present 5 times the cross-sectional area. The principal motivation for the original development of the single hole probe was that it does not involve the use of spatially separated measurement holes, which will cause a false measurement of the flow angle in flows where spatial gradients of velocity magnitude are encountered. In spite of this important advantage, they don't appear to have any development or application of the probe aside from Chu's. It seems probable of mechanical complexity involved in the mechanisms requires a roll positioning and cover extension has discouraged further development. The pressure response is also less sensitive with the single hole probe for a given cut angle as compared to that of a single port on a five hole probe.

Rotatable single hole pressure probe is developed using a hallow shaft stepping motor for roll positioning is shown in fig.2.

The mechanical design of the roll drive, the issue of sealing and shape of the sliding cover and most recently is the pressure response of the angle-cut tube for incompressible flow of varying magnitude and direction. The probe's roll mechanism is mechanically simpler than Chu's and provides roll positioning in 15° increments. Chu's approach of calibrating and using the devices as it were a 5-hole probe was abandoned: it was found that the special symmetry of the device can be exploited to yield a calibration function with 6 degrees of freedom. The calibration of the rotatable probe's response in this manner does not require a pressure measured by a square-cut sliding cover. The square-cut tube pressure measurement is still required in these cases where the static pressure is to be determined and for these situations we have obtained additional measurement using a completely separate, non-rotatable, square-cut total pressure probe.

The application of the calibrated probe to measurement in a flow of unknown velocity magnitude and direction has also been re-examined. 

PROBE DESIGN:

The new response function has been incorporated into a new flexible, implicit approach to processing pressure measurements made at more than 4 successive roll angles if desired to improve accuracy.  Initial tests of the entire procedure employed a prototype 1.3 mm diameter, 45 degree-cut-angle probe without a sliding cover to measure the velocity profile in a two-dimensional turbulent boundary layer.  Roll positioning was accomplished by a sub miniature, hollow shaft stepping motor¾a huge improvement allowed by modern advancements in stepping motor technology.  In these tests, velocity magnitude agreed with that measured by a conventional square-cut total pressure probe to within 0.5% of the stream speed, and flow angle was within 1 degree of expected values. The present work was conceived to extend the evaluation to a three-dimensional flow with small-scale features including static pressure variation.

Theoretical analysis methods:

It would be advantageous if the calibration characteristics and response of a Single-hole probe could be determined by analytical procedures. In fact, there are methods that can help to face up the analysis of pressure probes from a theoretical viewpoint. Some of these methods are:

- The streamline projection method

- The potential flow solution

For trapezoidal or cobra shaped head probes, analytical procedures of any type are difficult. These complex geometries, characterized by abrupt changes in contour, are subject to flow separation and viscous effects that are not modelled by current computational techniques. Nevertheless, the streamline projection method is used in addition to the experimental research as well as the computational investigations. It will be shown that this simple method can easily predict the single-hole probe calibration coefficients, at least qualitatively.

For probes of easy contour geometry, (e.g., cylindrical probe head), the streamline projection method is valid, but a potential flow solution can also predict the pressure distribution and the corresponding calibration characteristics to a reasonable accuracy.

While the analytic relationships are valuable for characterisation of probe behaviour and as a guide to the functional form of calibration equations, it is unlikely that they are capable of replacing individual probe calibrations. This is due to both the limitations of the derivation as well as the manufacturing irregularities of the probes. Regardless of the accuracy of the theoretical derivations, the latter effects may always necessitate individual probe calibrations, particularly for small sized probes.

Measurement of data and development of an interpolation procedure for the data analysis become responsibility of the probe user.

CALIBRATION TECHNIQUES:

For both, calibration and application, the probe's reference line is defined by some consistent characteristic of the probe's geometry. In application, a reference direction obtained by placing P1 with P2 is not always meaningful, since initially a known flow direction would be required to relate the balanced condition to an absolute spatial reference.

The probe can be operated in two ways:

  1. Nulling technique
  2. Stationary method (non-nulling technique)

Both methods offer advantages and disadvantages. Due to space restrictions in turbo machinery applications and wind tunnel blockage, the probe is often required to be small, and the difficulty associated with traversing and data acquisition encountered when the probe is used in a nulling fashion, make a non-nulling method a better alternative.

Calibration functions are then used to find the flow angles. The stream static and total pressures - and, thus, Mach number - can be determined in similar manner.

NULLING TECHNIQUE:

The nulling technique is the most accurate but mechanically complex. It is the most simple in terms of data analysis, as well. The probe is mounted on a three degree of freedom traversing system and is oriented such that the X-axis is parallel to the flow (yaw and pitch angles are both zero). The center pressure tap measures the stagnation pressure P1 and the pressures in the two outer tubes are equal (P2=P3) and proportional to the static pressure. Finally, the probe position is noted and the flow direction is determined from a calibrated scale.

This nulling technique requires a very sophisticated traversing system and long data acquisition time, since the probe must be yawed at each measurement location until the two pressures are equal. This can take a long time, especially if the probe is small and has a slow time response.

If space limitations or other considerations make nulling techniques impractical, three-hole probes in a non-nulling mode can be employed for measurements in low speed, incompressible flows.

STATIONARY METHOD:

The stationary method or non-nulling technique tends to be less accurate but offers simplicity in installation. The latter characteristic is the most important in turbo machine applications.

It is performed by setting the probe at constant pitch and yaw values with respect to the test section. The three pressures are measured at each measurement location by traversing the probe over the flow field. From these three measured pressures, the direction and magnitude of the flow with respect to the X-axis of the pressure probe are determined.

Although elegant in its simplicity, this technique encounters singularity when calibration for large angle of yaw is sought. So it is restricted to lower flow angle ranges, preventing its use in highly 3D flows.

ENERGY BERNOULLI EQUATION:

The flow motion is known to be a function of several non-dimensional parameters. The most important ones in aerodynamics are:

Reynolds number:

Mach number:

Gases at low velocity - the calibration experiments are conducted at Ma < 0.2 - can be considered as essentially incompressible (constant-density) fluids. The analysis of the steady flow for this sort of fluid starts with the conservation of mass, momentum and energy.

For one-dimensional flow along a stream tube (Fig. 1), the mass conservation equation for steady flow has the form:

If it is further assumed that the flow is in viscid, the momentum equation is:

The cross section of the stream tube must be small in order to consider the local values of the pressure and velocity.

For the steady state, the last term in Eq. (4) drops out, and the equation can be integrated along the direction s of the stream tube, to result in the Bernoulli conservation equation for energy:

The pressure pin Eq. (5) is the static pressure. It is the component of the pressure that represents fluid hydrostatic effects. And in principle, it is measured by an instrument that moves along with the fluid. This is, however, inconvenient, and the pressure is usually measured via a small hole in a wall arranged so that it does not disturb the flow. The quantity is usually called dynamic pressure. It is the component of the fluid that represents fluid kinetic energy.

Total pressure pt, sometimes also called “stagnation pressure”, is defined as the pressure that would be reached if the local flow is imagined to slow down to zero velocity, frictionless. Total pressure is the sum of static and dynamic pressure:

From measurements of the total and static pressures, the velocity can be obtained as,

This follows readily from Eq

For Eq. (6) and (7) to apply, the probe must not disturb the flow, and it must be carefully aligned parallel to the stream.      

DEFINITION OF HOLE AND CALIBRATION COEFFICIENTS:

The pressure sensed by the hole idiffer from the free stream static pressure p. The hole coefficient ki is usually used in the following form:

Where irepresents number of holes.

It is apparent that the calibration characteristics must include data that represent pressure differences in the yaw plane as well as differences between the measured and the true, local total and static pressures. When the probe is used to measure these quantities, the relationship between them and the yaw angle Δβ is described by the calibration coefficients. These pressure coefficients must be defined so that they are independent of velocity and are a function only of the flow angularity.

Total pressure coefficient:

Static pressure coefficient:

EFFECT OF TURBULENCE ON PRESSURE PROBE:

The single-hole yaw meter:

This device is a single chamfered tube over which can be slid a sleeve which, in its forward position, converts the device into a pitot tube. The pitot pressure having been measured, yaw and pitch angles can be deduced from readings of the single chamfered tube at, at most, four azimuthal positions. The yaw calibration in uniform flow is shown in figure 13(a), and, unlike the conventional three-hole yaw meter but like the Gupta yaw meter, breaks downward from a linear variation at a yaw angle of about 15". In turbulent flow, however, the yaw calibration breaks upwards from the linear portion at an angle of about 20". The slope of the linear portion is about 2 % greater in turbulent flow than in uniform flow.

PRESSURE PROBE MEASUREMENTS:

The standard method for measuring mean and average velocity is to use a pitot static tube. Accurate use of this probe relies on aligning the probe to known primary flow direction. Chu, Bryer and Pankhurst provide a good summary of the accuracy penalties incurred for misalignment for different Pitot - static tube designs. For complex flows, both dynamic and static pressures are required as well as flow direction. This is most commonly achieved by using combination or multi-hole pressure probe which typically requires calibration in known flow conditions representative of those in which they will be used.

Many different types of probes have been proposed in the past, either for measuring flow direction, flow speed or both. In accuracies due to Reynolds-number effects can be significant and need to be considered for accurate measurements. Westphal developed a single hole probe in which a single tube cut at 45° is rotated about its axis by a miniature stepper motor. This allows velocity, magnitude and angle measurements with a significantly smaller probe, although the measurement procedure.

When using multi-hole probes, one issue which should always be kept in mind is that ultimate accuracy of the pressure probe relies on the accuracy of the calibration. Moreover, one uncertainty that remains with most exiting calibration techniques is that the calibrations are performed at steady pitch and yaw angles in uniform, and no account of the nonlinear effects of the turbulent flow are accounted. The level of this uncertainty remains unclear.

In order to use a single hole pressure probe in a ``non-nulling'' mode, it is necessary to find a relationship between the angle and the velocity of the flow, the static pressure and the pressures measured in the probe holes. Such a relationship is obtained by means of a direct calibration of the probe. Practically, all the methods that can be found in the literature regarding the calibration procedure of multi hole pressure probes [8] have a common feature: they use normalized calibration coefficients. Such coefficients, as defined in [9], are obtained for a three-hole probe as a function of the total pressure, P0, the static pressure, PS , and all the pressures measured in the central hole, PC , and in both left-hand side, PL, and right-hand side holes, PR, yielding:

Where Cα represents the angular coefficient, CP0 and CPs are the total and static pressure coefficients and Q is the normalization factor that is introduced in all the coefficients. All the coefficients become independent of the dynamic pressure if normalized with the factor Q. Thus, the angular coefficient is only a function of the flow angle, so both static and total pressure coefficients provide the velocity magnitude and the static pressure in the flow. The calibration setup of a probe is based on a repositioning sequence of the probe inside a known flow field. This means that the relative angle position of the probe respect to the flow direction has to be progressively modified using a highly-precise angular stepping mechanism. For every angular position, the total and static pressures as well as all the pressure measurements in the holes are stored. Using these pressure values, all the calibration coefficients defined in (1) are directly obtained. Since the probe is symmetric, it is possible to fulfill the calibration just considering positive (or negative) incidence angles of the flow. However, in order to avoid slight differences derived from imperfections of the probe during the manufacturing process, it is preferred to conduct a complete calibration setup. The calibration coefficients to be obtained following (1) and the procedure to determine both velocity magnitude and direction of the flow after completing a measuring sequence with a CTHP is sketched in Fig. 3. The calibration provides the variations of the coefficients Cα, CP0 and CPs with the flow angle α. Then, the angular coefficient is obtained from the different pressures sensed in the holes when measuring. Once the angular coefficient Cα is calculated, the flow angle is known and the values of both pressure coefficients, CP0 and CPs are also determined. These coefficients allow us to calculate both static and total pressures of the flow. The difference between them, i.e. the dynamic pressure, is immediately related to the velocity magnitude of the flow.

The main goal is placed on the development of general relationships to reduce the numerical calculations that are necessary in the post-processing. However, due to the improvement of the computational resources in recent years, nowadays it is better to employ interpolation.

Typical problem that arises in the normalization with the factor Q is the introduction of singular points for all calibration coefficients when Q turns to zero. In fact, the problem is not really associated to the appearance of singular points, which could be avoided easily by means of some kind of mathematical operator. The definitive problem lies in the non-monotonous behaviour of the curve of the angular coefficient when crossing from one side to the other side of the singularities. The second limiting factor of the CTHP performance refers to the velocity range that is measurable. Basically, the calibration method is based on the fact that when the calibration coefficients are normalized, they vary significantly with the flow incidence angle, but are practically independent of the flow velocity [9]. The normalized calibration coefficients are obtained considering that the pressure distribution around a cylinder can be expressed as follows:

Where

Pd corresponds to the dynamic pressure

Cp is the pressure coefficient.

θ is the angle between every point in the cylinder surface and the free stream direction. 

In the case of single hole pressure probe, with construction angle δ, the pressure in hole at different angles is given by:

If all expression above is introduced in the definition of the normalized calibration coefficients, it is easily demonstrated that the coefficient are independent of both total and static pressure, being determined only by the pressure coefficient Cp.

ANGULAR CALIBRATION:

This section contains a theoretical analysis of the calibration coefficients in the case of a CTHP (which is similar to single hole pressure probe) with an angular distance of 45 degrees between the holes. The calibration coefficients that are obtained using a traditional normalizing factor Q are compared to the new set of coefficients derived from an improved normalization factor QN. The distribution of the pressure coefficient Cp in the central hole should be obtained rolling the probe 360 degrees about its own axis in a flow field. However, since the interest is now placed in developing a theoretical framework of the calibration setup, the analysis is afforded using bibliographic data. Thus, we have employed for convenience a curve fitting tool based on splines through the experimental data collected in [16] for a Reynolds  number of 2.3Ã-104. The Cp distributions in both left (L) and right (R) holes have been constructed shifting the original data for the central hole ±45° respectively. The final distributions in the three-holes, which are a function of the flow angle α, are plotted in Fig. 7. Though adapted from experimental results, Cp distributions imposed like this have to be considered as ideal, since these assumptions imply that there is no uncertainty associated either to the angular distance of the holes or to the probe misalignment. As expected, the maximum of the pressure coefficient in the central port is obtained for a flow angle of 0 degrees. Similarly, maximum values for both left and right ports are reached respectively at ±45°. If the construction angle of the probe would be 60 degrees, then the maximum values for both lateral holes would be placed at ±60°. Following, with the distributions of the pressure coefficient of Fig. 7, the calibration coefficients that are derived from the traditional calibration (normalization factor Q) are shown in Fig. 8. The total and static pressure coefficients, CP0 and CPs , are very similar. Both of them are symmetric respect to an incidence flow angle of 0 degrees. In the angular range of ±30° are positive, showing values between 0 and 4. For the angular range of ±10° are nearly constant, rising as the flow angle increases towards ±30°. On the contrary, the angular coefficient C_ is anti-symmetric respect to a flow angle of 0 degrees. This way, it takes positive values for negative incidence angles and negative values for positive incidence angles, ranged from -6 to 6. At a 0 degree flow angle, the angular coefficient is zero. The significance of Fig. 8 is that the angular range of a CTHP calibrated with the traditional method is just limited to the interval between -30 and 30 degrees. Outside of this angular range, the angular coefficient is no longer monotonous with the flow angle, as illustrated in Fig. 9. Shown in the figure is the distribution of the angular coefficient calculated with the regular normalization factor Q, for two probes with construction angles of 45 (black line) and 60 degrees (gray line). The x-axis has been extended to include an angular range of ±70°.

Notice that outside the interval of [-30°, +30°], singular points are introducing sharp discontinuities in the coefficient. As a result, beyond the internal angular range, the calibration data cannot be reduced to obtain a unique flow angle, limiting the use of the probe. In addition, as observed in Fig. 9, both probes with different construction angles exhibit a similar attainable angular range. As a consequence of this result, an attempt was made to bring out a new method to calibrate a CTHP, with the final objective of increasing the angular range of the probe. By definition, the angular coefficient is calculated as the pressure differential on the left and right holes, divided by the normalization factor Eq. (1). If the normalization factor would be value unity, the pressure differential between ports L and R would offer a monotonous angular coefficient along an angular range of, at least, twice the construction angle of the probe (see Fig. 7). With that pressure differential in the numerator of the angular coefficient, it was necessary to find a normalization factor as constant as possible when varying the flow angle. This new normalization factor is labelled as QN.

The pressure distribution in the hole is quite similar to the inviscid behaviour of potential flow (see Fig. 4), so it could be expressed as a function of cos2α. The pressure distribution in R is obtained moving the central distribution an angular distance equal to the construction angle of the probe. As a consequence, it could be expressed as a function of sin2α. However, the objective is to find an approximate relationship between the pressure ports, rather than an exact mathematical expression. Then, since cos2α + sin2α=1, the new normalization factor QN could be defined as the sum of both pressures in C and R. On the other hand, QN must be independent of the static pressure. Considering that the pressure in L is nearly constant for positive flow angles, it is feasible to subtract twice the sum of the pressures in the central and right holes in order to eliminate the static pressure in the definition of QN.

This definition provides a normalization factor, independent of the static pressure, which can be expressed as the product of the dynamic pressure times a certain function of the flow angle. Thus, by means of QN, an angular coefficient independent of the dynamic pressure and only function of the flow angle is available. Previous discussion is exclusively applicable for positive flow angles. In the case of negative incidences, it is necessary to reconsider this reasoning, so it can be deduced that the normalization factor should be now

In the calibration process, it is possible to define different angular coefficients in case of positive or negative flow angles, but when measuring this discrimination is unpractical. For a real measurement, you may have different angular coefficients only if you can differentiate the angular range using some of the pressure values sensed in the probe holes. Fortunately, in the case of the normalization factor QN defined through (5) and (6), the flow angles are positive when PR > PL and negative when PL > PR. Even when the flow angle is zero, both pressures in the left and right holes are the same, so QN presents no discontinuities for α=0°. In summary, the normalization factor QN is defined as:

UNCERTAINTY:

Having defined this new calibration method, it is necessary to analyze the uncertainty levels associated to the normalization factor QN . Present investigations confirm that not only the angular range is increased, but also the uncertainty levels are roughly different respect to typical uncertainty levels associated to the traditional calibration. In many ways, the factor QN can be considered as two times the traditional factor Q, since it is calculated using twice the pressure values of the holes. Then, the uncertainty of the measurements associated to the factor QN may be estimated somehow as twice the uncertainty of the measurements obtained using Q. Nevertheless, the uncertainty in the angular coefficient is small because of the reduced transfer of uncertainty in the mathematical process. In fact, though the uncertainty in the flow angle is higher (shown later) for small flow angles, the uncertainty in the velocity magnitude and the static pressure are lower, even inside the typical angular range of traditional calibration (between ±10° and ±30°). The methodology proposed has been followed to evaluate the uncertainty of the flow variables measured with a single hole pressure probe. According to previous distributions of the angular coefficient, the flow angle can be expressed in terms of a unique explicit analytic function of Cα. Hence, the uncertainty of the flow angle is:

Using the definition of the angular coefficient in (1), the uncertainty of Cα can be also obtained following an analogous algebra:

Where Q is the traditional normalization factor. To that end, it has been supposed that the uncertainty in the pressure is the same for all the holes, being equal to the uncertainty in a pressure measurement, IP, which - in a typical experiment - is basically determined by the uncertainty of the pressure transducers.

The uncertainty of the traditional normalization factor Q is given by:

Substituting Eq. (10) into (9), the uncertainty of the angular coefficient is expressed as a function of the uncertainty in the pressure measurement:

Finally, including Eq. (11) in (8), the uncertainty of the flow angle is:

Where the derivative of the angle with respect to the angular coefficient must be evaluated numerically. Analogous considerations lead to define the relative uncertainty for the velocity magnitude as well as the static pressure uncertainty, according to:

Applying identical deductions in the case of the new calibration method, the uncertainty of the normalization factor QN is:

From (15), the uncertainty associated to the flow angle, the relative uncertainty for the velocity magnitude and the static pressure uncertainty for the new calibration are given by:

The right column of Fig. 12 shows the distribution of all the uncertainties formulated before as a function of the flow angle. Both results for traditional (dashed lines) and new (solid lines) calibrations are included in the array of plots. In addition, the left column of the figure reproduces the calibration coefficients that are derived from both normalization factors Q and QN. All the uncertainties have been made non-dimensional. The uncertainty of the flow angle is expressed as a percentage of the uncertainty in the pressure measurement, IP, relative to the dynamic pressure Pd. This means that, for instance, if α=30° the uncertainty in the flow angle is about 0.5 degrees for every 1% of IP=Pd. The uncertainty for the static pressure is referenced to the uncertainty in the pressure measurement. Finally, the relative uncertainty for the velocity has

been made non-dimensional with the uncertainty in the pressure measurement relative to the dynamic pressure, instead of using the static pressure. This implies that the static pressure of the flow cannot be excessively high with respect to the dynamic pressure.

From Fig. 12, it is observed a common feature for both calibrations: the uncertainty of the flow angle tends to infinite when the normalization factor tends to zero (at ±30° for Q and ±70° for QN). This is because the angle uncertainty is calculated dividing by the normalization factor Eq . (12) and (16). Alternatively, though the relative uncertainty of the velocity and the uncertainty of the static pressure are not obtained dividing by the normalization factor Eq. (13), (14), (17) and (18) _, both variables tend to infinite at ±30° when using the traditional calibration. This is a consequence of the behaviour of both static and total pressure coefficients, which are also infinite at ±30° (left column of the figure). On the contrary, the relative uncertainty of the velocity and the uncertainty of the static pressure in case of the new calibration are increased towards ±70°, but limited to finite values. For a flow angle of zero degrees, the uncertainty in the angle measurement is the same for both calibrations.

Inside the angular range of ±30°, the uncertainty Iα is slightly higher when using the new calibration than the traditional one. Even so, at ±60° it is considerable small with just 1 degree for every 1% of IP=Pd. The relative uncertainty in the velocity magnitude and the uncertainty in the static pressure are a bit higher when calibrating with QN for α = 0°. On the other hand, in the angular intervals [-30°, -10°] and [10°, 30°], the results show a better performance of the new calibration method. Complementarily, the relative uncertainty of velocity takes values ranged between 0.5 and 1.2 times IP=Pd for the whole angular range, which is an exceptional good result. The uncertainty of the static pressure is strongly increased from 1.2 to 2 times IP at ±40°, reaching up to 6 times IP when being nearby ±70°. Anyway, overall differences between the uncertainties evaluated for both calibration methods are not significant in the angular range of ±30°. Therefore, the new calibration is not only providing a CTHP that is capable of measuring angular variations of the flow up to 140 degrees, but it also presents reasonable low uncertainties for the whole angular range of the probe (±70°).

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